Ratios, Proportions, and Similarity

Ratios, Proportions, and Similarity
L21_Ratios and Proportions Ratios, Proportions, and Similarity Eleanor Roosevelt High School Chin-Sung Lin

L21_Ratios and Proportions
ERHS Math Geometry Ratio and Proportion Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Ratio A ratio is a comparison by division of two quantities that have the same units of measurement The ratio of two numbers, a and b, where b is not zero, is the number a/b e.g. AB = 4 cm and CD = 5 cm AB cm CD cm * A ratio has no units of measurement = = or 4 to 5 or 4 : 5 Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Ratio Since a ratio, like a fraction, is a comparison of two numbers by division, a ratio can be simplified by dividing each term of the ratio by a common factor e.g. AB = 20 cm and CD = 5 cm AB cm CD cm * A ratio is in simplest form (or lowest terms) when the terms of the ratio have no common factor greater than 1 = = or 4 to 1 or 4 : 1 Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Ratio A ratio can also be used to express the relationship among three or more numbers e.g. the measures of the angles of a triangle are 45, 60, and 75, the ratio of these measures can be written as 45 : 60 : or, in lowest terms, ? Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Ratio A ratio can also be used to express the relationship among three or more numbers e.g. the measures of the angles of a triangle are 45, 60, and 75, the ratio of these measures can be written as 45 : 60 : or, in lowest terms, 3 : 4 : 5 Mr. Chin-Sung Lin

ERHS Math Geometry Ratio Example If the lengths of the sides of a triangle are in the ratio 3 : 3 : 4, and the perimeter of the triangle is 120 cm, Find the lengths of the sides Mr. Chin-Sung Lin

ERHS Math Geometry Ratio Example If the lengths of the sides of a triangle are in the ratio 3 : 3 : 4, and the perimeter of the triangle is 120 cm, Find the lengths of the sides x: the greatest common factor three sides: 3x, 3x, and 4x 3x + 3x + 4x = 120 10x = 120, x = 12 The measures of the sides: 3(12), 3(12), and 4(12) or 36 cm, 36 cm, and 48 cm Mr. Chin-Sung Lin

ERHS Math Geometry Definition of Rate A rate is a comparison by division of two quantities that have different units of measurement e.g. A UFO moves 400 m in 2 seconds the rate of distance per second (speed) is distance m time s * A rate has unit of measurement 200 m/s = = Mr. Chin-Sung Lin

Definition of Proportion
L21_Ratios and Proportions ERHS Math Geometry Definition of Proportion A proportion is a statement that two ratios are equal. It can be read as “a is to b as c is to d” a c b d * When three or more ratios are equal, we can write and extended proportion a c e g b d f h = or a : b = c : d = = = Mr. Chin-Sung Lin

Definition of Extremes and Means
L21_Ratios and Proportions ERHS Math Geometry Definition of Extremes and Means A proportion a : b = c : d a : b = c : d extremes means Mr. Chin-Sung Lin

Definition of Constant of Proportionality
L21_Ratios and Proportions ERHS Math Geometry Definition of Constant of Proportionality When x is proportional to y and y = kx, k is called the constant of proportionality y x k = is the constant of proportionality Mr. Chin-Sung Lin

Properties of Proportions
L21_Ratios and Proportions ERHS Math Geometry Properties of Proportions Mr. Chin-Sung Lin

Cross-Product Property
L21_Ratios and Proportions ERHS Math Geometry Cross-Product Property In a proportion, the product of the extremes is equal to the product of the means a c b d b ≠ 0, d ≠ 0 * The terms b and c of the proportion are called means, and the terms a and d are the extremes then a x d = b x c = Mr. Chin-Sung Lin

Interchange Extremes Property
L21_Ratios and Proportions ERHS Math Geometry Interchange Extremes Property In a proportion, the extremes may be interchanged a c d c b d b a b ≠ 0, d ≠ 0, a ≠ 0 = then = Mr. Chin-Sung Lin

Interchange Means Property (Alternation Property)
L21_Ratios and Proportions ERHS Math Geometry Interchange Means Property (Alternation Property) In a proportion, the means may be interchanged a c a b b d c d b ≠ 0, d ≠ 0, c ≠ 0 = then = Mr. Chin-Sung Lin

ERHS Math Geometry Inversion Property The proportions are equivalent when inverse both ratios a c b d b d a c b ≠ 0, d ≠ 0, a ≠ 0, c ≠ 0 then = = Mr. Chin-Sung Lin

Equal Factor Products Property
L21_Ratios and Proportions ERHS Math Geometry Equal Factor Products Property If the products of two pairs of factors are equal, the factors of one pair can be the means and the factors of the other the extremes of a proportion c a b d b ≠ 0, d ≠ 0 a, b are the means, and c, d are the extremes a x b = c x d then = Mr. Chin-Sung Lin

ERHS Math Geometry Composition Property The proportions are equivalent when adding unity to both ratios a c a + b c + d b d b d b ≠ 0, d ≠ 0 then = = Mr. Chin-Sung Lin

ERHS Math Geometry Division Property The proportions are equivalent when subtracting unity to both ratios a c a - b c - d b d b d b ≠ 0, d ≠ 0 then = = Mr. Chin-Sung Lin

Definition of Geometric Mean (Mean Proportional)
L21_Ratios and Proportions ERHS Math Geometry Definition of Geometric Mean (Mean Proportional) Suppose a, x, and d are positive real numbers a x x d Then, x is called the geometric mean or mean proportional, between a and d = then x2 = a d or x = √ad Mr. Chin-Sung Lin

ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

Example - Ratios and Rates
L21_Ratios and Proportions ERHS Math Geometry Example - Ratios and Rates Maria has two job opportunities. If she works for a healthcare supplies store, she will be paid $60 daily by working 5 hours a day. If she works for a grocery store, she will be paid $320 weekly by working 8 hours a day and 5 days per week What is the pay rate for each job? What is the ratio between the pay rates of healthcare supplies store and grocery store? Mr. Chin-Sung Lin

Example - Ratios and Rates
L21_Ratios and Proportions ERHS Math Geometry Example - Ratios and Rates Maria has two job opportunities. If she works for a healthcare supplies store, she will be paid $60 daily by working 5 hours a day. If she works for a grocery store, she will be paid $320 weekly by working 8 hours a day and 5 days per week What is the pay rate for each job? Healthcare: $12/hr, Grocery: $8/hr What is the ratio between the pay rates of healthcare supplies store and grocery store? Healthcare:Grocery = 12:8 = 3:2 Mr. Chin-Sung Lin

ERHS Math Geometry Example - Ratios The perimeter of a rectangle is 48 cm. If the length and width of the rectangle are in the ratio of 2 to 1 What is the length of the rectangle? Mr. Chin-Sung Lin

ERHS Math Geometry Example - Ratios The perimeter of a rectangle is 48 cm. If the length and width of the rectangle are in the ratio of 2 to 1 What is the length of the rectangle? Width: x Length: 2x 2 (2x + x) = 48 3x = 24, x = 8 2x = 16 Length is 16 cm Mr. Chin-Sung Lin

ERHS Math Geometry Example - Ratios The measures of an exterior angle of a triangle and the adjacent interior angle are in the ratio 7 : 3. Find the measure of the exterior angle Mr. Chin-Sung Lin

ERHS Math Geometry Example - Ratios The measures of an exterior angle of a triangle and the adjacent interior angle are in the ratio 7 : 3. Find the measure of the exterior angle Exterior angle: 7x Interior angle: 3x 7x + 3x = 180 10x = 180, x = 18 7x = 126 Measure of the exterior angle is 126 Mr. Chin-Sung Lin

ERHS Math Geometry Example - Proportions Solve the proportions for x x + 2 x - 1 = Mr. Chin-Sung Lin

ERHS Math Geometry Example - Proportions Solve the proportions for x x + 2 x - 1 2 ( 2x – 1) = 5 (x + 2) 4x – 2 = 5x + 10 -12 = x x = -12 = Mr. Chin-Sung Lin

Example - Geometric Mean
L21_Ratios and Proportions ERHS Math Geometry Example - Geometric Mean If the geometric mean between x and 4x is 8, solve for x Mr. Chin-Sung Lin

L21_Ratios and Proportions ERHS Math Geometry Example - Geometric Mean If the geometric mean between x and 4x is 8, solve for x x (4x) = 82 4x2 = 64 x2 = 16 x = 4 Mr. Chin-Sung Lin

Example - Properties of Proportions
L21_Ratios and Proportions ERHS Math Geometry Example - Properties of Proportions 4x y2 8x2 = 45 y2 4 3 15y x2 5 4x 2x y2 If = , which of the following statements are true? Why? (x ≠ 0, y ≠ 0) 4x + 9y 5y + 2x 9y x 9y x 4x - 9y y - 2x = = = = Mr. Chin-Sung Lin

ERHS Math Geometry Midsegment Theorem Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Midsegment Theorem A line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB C E D A B Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Midsegment Theorem A line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side Given: ∆ABC, D is the midpoint of AC, and E is the midpoint of BC Prove: DE || AB, and DE = ½ AB C E D F A B Mr. Chin-Sung Lin

Divided Proportionally Theorem
L21_Ratios and Proportions ERHS Math Geometry Divided Proportionally Theorem Mr. Chin-Sung Lin

Definition of Divided Proportionally
L18_Trapezoids ERHS Math Geometry Definition of Divided Proportionally Two line segments are divided proportionally when the ratio of the lengths of the parts of one segment is equal to the ratio of the lengths of the parts of the other e.g., in ∆ABC, DC/AD = 2/1 EC/BE = 2/1 then, the points D and E divide AC and BC proportionally C A B E D Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Divided Proportionally Theorem If two line segments are divided proportionally, then the ratio of the length of a part of one segment to the length of the whole is equal to the ratio of the corresponding lengths of the other segment Given: In ∆ABC, AD/DB = AE/EC Prove: AD/AB = AE/AC A B C E D Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Divided Proportionally Theorem D E A B C Statements Reasons 1. AD/DB = AE/EC Given 2. DB/AD = EC/AE Inversion property 3. (DB+AD)/AD = (EC+AE)/AE 3. Composition property 4. DB+AD = AB, EC+AE = AC 4. Partition postulate 5. AB/AD = AC/AE 5. Substitution postulate 6. AD/AB = AE/AC Inversion property Mr. Chin-Sung Lin

Converse of Divided Proportionally Theorem
L18_Trapezoids ERHS Math Geometry Converse of Divided Proportionally Theorem If the ratio of the length of a part of one line segment to the length of the whole is equal to the ratio of the corresponding lengths of another line segment, then the two segments are divided proportionally Given: In ∆ABC, AD/AB = AE/AC Prove: AD/DB = AE/EC A B C E D Mr. Chin-Sung Lin

Converse of Divided Proportionally Theorem
L22_Similar Triangles ERHS Math Geometry Converse of Divided Proportionally Theorem D E A B C Statements Reasons 1. AD/AB = AE/AC Given 2. AB/AD = AC/AE Inversion property 3. (AB-AD)/AD = (AC-AE)/AE 3. Division property 4. AB-AD = DB, AC-AE = EC 4. Partition postulate 5. DB/AD = EC/AE 5. Substitution postulate 6. AD/DB = AE/EC Inversion property Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Divided Proportionally Theorem & Converse of Divided Proportionally Theorem Two line segments are divided proportionally if and only if the ratio of the length of a part of one segment to the length of the whole is equal to the ratio of the corresponding lengths of the other segment A B C E D Mr. Chin-Sung Lin

Application Examples ERHS Math Geometry Mr. Chin-Sung Lin
L22_Similar Triangles ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

Example - Divided Proportionally
L18_Trapezoids ERHS Math Geometry Example - Divided Proportionally In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. BC = 7x+5, DE = 4x-2, BD = 2x+1, AC = 9x+1 Find DE, BC, BD, AB, AC, and AE A B C E D Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Example - Divided Proportionally In ∆ABC, D is the midpoint of AB and E is the midpoint of AC. BC = 7x+5, DE = 4x-2, BD = 2x+1, AC = 9x+1 Find DE, BC, BD, AB, AC, and AE 7x + 5 = 2 (4x – 2) 7x + 5 = 8x – 4 x = 9 DE = 4 (9) – 2 = 34 BC = 2 (34) = 68 BD = 2 (9) + 1 = 19 AB = 2 (19) = 38 AC = 9 (9) + 1 = 82 AE = 82/2 = 41 A B C E D Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Example - Divided Proportionally ABC and DEF are line segments. If AB = 10, AC = 15, DE = 8, and DF = 12,do B and E divide ABC and DEF proportionally? Mr. Chin-Sung Lin

L18_Trapezoids ERHS Math Geometry Example - Divided Proportionally ABC and DEF are line segments. If AB = 10, AC = 15, DE = 8, and DF = 12,do B and E divide ABC and DEF proportionally? BC = 15 – 10 = 5 EF = 12 – 8 = 4 AB : BC = 10 : 5 = 2 : 1 DE : EF = 8 : 4 = 2 : 1 AB : BC = DE : EF So, B and E divide ABC and DEF proportionally Mr. Chin-Sung Lin

Similar Triangles ERHS Math Geometry Mr. Chin-Sung Lin
L22_Similar Triangles ERHS Math Geometry Similar Triangles Mr. Chin-Sung Lin

Definition of Similar Figures
L22_Similar Triangles ERHS Math Geometry Definition of Similar Figures Two figures that have the same shape but not necessarily the same size are called similar figures Mr. Chin-Sung Lin

Definition of Similar Polygons
L22_Similar Triangles ERHS Math Geometry Definition of Similar Polygons Two (convex) polygons are similar (~) if their consecutive vertices can be paired so that: Corresponding angles are congruent The lengths of corresponding sides are proportional (have the same ratio, called ratio of similitude) Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Definition of Similar Polygons Both conditions must be true for polygons to be similar: Corresponding angles are congruent The lengths of corresponding sides are proportional corresponding angles are congruent P Q S R 6 9 60o A B C D 4 6 W X Z Y 6 12 60o corresponding sides are proportional Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Definition of Similar Polygons If two polygons are similar, then their corresponding angles are congruent and their corresponding sides are in proportion and If two polygons have corresponding angles that are congruent and corresponding sides that are in proportion, then the polygons are similar Mr. Chin-Sung Lin

Definition of Similar Triangles
L22_Similar Triangles ERHS Math Geometry Definition of Similar Triangles Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional (The number represented by the ratio of similitude is called the constant of proportionality) Mr. Chin-Sung Lin

Example of Similar Triangles
L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles A  X, B  Y, C  Z AB = 6, BC = 8, and CA = 10 XY = 3, YZ = 4 and ZX = 5 Show that ABC~XYZ A B C 6 8 10 X Y Z 3 4 5 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles A  X, B  Y, C  Z AB BC CA XY YZ ZX Therefore ABC~XYZ = = = A B C 6 8 10 X Y Z 3 4 5 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles The sides of a triangle have lengths 4, 6, and 8. Find the sides of a larger similar triangle if the constant of proportionality is 5/2 ? 4 6 8 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles Assume x, y, and z are the sides of the larger triangle, then x y z = = = 4 6 8 y = 20 x = 10 z = 15 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles In ABC, AB = 9, BC = 15, AC = 18. If ABC~XYZ, and XZ = 12, find XY and YZ 9 15 18 A B C X Y Z ? 12 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Example of Similar Triangles Since ABC~XYZ, and XZ = 12, then XY YZ = = 9 15 18 A B C X Y Z 6 10 12 Mr. Chin-Sung Lin

Equivalence Relation of Similarity
L22_Similar Triangles ERHS Math Geometry Equivalence Relation of Similarity Mr. Chin-Sung Lin

Reflexive Property Any geometric figure is similar to itself ABC~ABC
L22_Similar Triangles ERHS Math Geometry Reflexive Property Any geometric figure is similar to itself ABC~ABC Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Symmetric Property A similarity between two geometric figures may be expressed in either order If ABC~DEF, then DEF~ABC Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Transitive Property Two geometric figures similar to the same geometric figure are similar to each other If ABC~DEF, and DEF~RST, then ABC~RST Mr. Chin-Sung Lin

Postulates & Theorems ERHS Math Geometry Mr. Chin-Sung Lin
L22_Similar Triangles ERHS Math Geometry Postulates & Theorems Mr. Chin-Sung Lin

Postulate of Similarity
L22_Similar Triangles ERHS Math Geometry Postulate of Similarity For any given triangle there exists a similar triangle with any given ratio of similitude Mr. Chin-Sung Lin

Angle-Angle Similarity Theorem (AA~)
L22_Similar Triangles ERHS Math Geometry Angle-Angle Similarity Theorem (AA~) If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar Given: ABC and XYZ with A  X, and C  Z Prove: ABC~XYZ X Y Z A B C Mr. Chin-Sung Lin

Angle-Angle Similarity Theorem (AA~)
L22_Similar Triangles ERHS Math Geometry Angle-Angle Similarity Theorem (AA~) Statements Reasons 1. A  X, and C  Z 1. Given 2. Draw RST ~ ABC Postulate of similarity with RT/AC = XZ/AC 3. R  A, T  C 3. Definition of similar triangles 4. R  X, T  Z Transitive property 5. AC =AC Reflecxive postulate 6. RT = XZ Multiplication postulate 7. RST  XYZ 7. SAS postulate 8. RST ~ XYZ 8. Congruent ’s are similar ’s 9. ABC ~ XYZ 9. Transitive property of similarity A B C R S T X Y Z

Example of AA Similarity Theorem
L22_Similar Triangles ERHS Math Geometry Example of AA Similarity Theorem Given: mA = 45 and mD = 45 Prove: ABC~DEC 45o A B C D E Mr. Chin-Sung Lin

Example of AA Similarity Theorem
L22_Similar Triangles ERHS Math Geometry Example of AA Similarity Theorem 45o A B C D E Statements Reasons 1. mA = 45 and mD = Given 2. A  D Substitution property 3. ACB  DCE 3. Vertical angles 4. ABC~DEC AA similarity theorem Mr. Chin-Sung Lin

Side-Side-Side Similarity Theorem (SSS~)
L22_Similar Triangles ERHS Math Geometry Side-Side-Side Similarity Theorem (SSS~) Two triangles are similar if the three ratios of corresponding sides are equal Given: AB/XY = AC/XZ = BC/YZ Prove: ABC~XYZ A B C X Y Z Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Side-Side-Side Similarity Theorem (SSS~) Two triangles are similar if the three ratios of corresponding sides are equal Given: AB/XY = AC/XZ = BC/YZ Prove: ABC~XYZ X Y Z A B C D E Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Side-Side-Side Similarity Theorem (SSS~) Statements Reasons 1. AB/XY = AC/XZ = BC/YZ 1. Given 2. Draw DE, D is on AB, E is on AC 2. Postulate of similarity with AD = XY, DE || BC 3. ADE  B, and AED  C 3. Corresponding angles 4. ADE ~ ABC 4. AA similaity theorem 5. AB/AD = AC/AE = BC/DE 5. Corresponding sides proportional 6. AB/XY = AC/AE = BC/DE 6. Substitutin postulate 7. AC/AE = AC/XZ, BC/DE = BC/YZ 7. Transitive postulate 8. (AC)(XZ) = (AC)(AE) 8. Cross product (BC)(DE) = (BC)(YZ) 9. AE = XZ, DE = YZ 9. Division postulate 10. ADE  XYZ SSS postulate 11. ADE ~ XYZ Congruent ’s are similar ’s 12. ABC ~ XYZ Transitive property of similarity X Y Z A B C D E

Side-Angle-Side Similarity Theorem (SAS~)
L22_Similar Triangles ERHS Math Geometry Side-Angle-Side Similarity Theorem (SAS~) Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent Given: A  X, AB/XY = AC/XZ Prove: ABC~XYZ A B C X Y Z Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Side-Angle-Side Similarity Theorem (SAS~) Two triangles are similar if the ratios of two pairs of corresponding sides are equal and the corresponding angles included between these sides are congruent Given: A  X, AB/XY = AC/XZ Prove: ABC~XYZ X Y Z A B C D E Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Side-Angle-Side Similarity Theorem (SAS~) Statements Reasons 1. A  X, AB/XY = AC/XZ 1. Given 2. Draw DE, D is on AB, E is on AC 2. Postulate of similarity with AD = XY, DE || BC 3. ADE  B, and AED  C 3. Corresponding angles 4. ADE ~ ABC 4. AA similaity theorem 5. AB/AD = AC/AE Corresponding sides proportional 6. AB/XY = AC/AE Substitutin postulate 7. AC/AE = AC/XZ Transitive postulate 8. (AC)(XZ) = (AC)(AE) 8. Cross product 9. AE = XZ Division postulate 10. ADE  XYZ SAS postulate 11. ADE ~ XYZ Congruent ’s are similar ’s 12. ABC ~ XYZ Transitive property of similarity X Y Z A B C D E

Example of SAS Similarity Theorem
L22_Similar Triangles ERHS Math Geometry Example of SAS Similarity Theorem Prove: ABC~DEC Calculate: DE 16 A B C D E 10 12 8 6 ? Mr. Chin-Sung Lin

Example of SAS Similarity Theorem
L22_Similar Triangles ERHS Math Geometry Example of SAS Similarity Theorem Prove: ABC~DEC Calculate: DE 16 A B C D E 10 12 8 6 5 Mr. Chin-Sung Lin

Triangle Proportionality Theorem (Side-Splitter Theorem)
L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem (Side-Splitter Theorem) Mr. Chin-Sung Lin

Triangle Proportionality Theorem
L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally Given: DE || BC Prove: AD AE DB EC D E A B C = Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem D E A B C Statements Reasons 1. BC || DE Given 2. A  A Reflexive property 3. ABC  ADE 3. Corresponding angles 4. ABC~ADE AA similarity theorem 5. AB/AD = AC/AE 5. Corresp. sides proportional 6. (AB-AD)/AD = (AC-AE)/AE 6. Proportional by division 7. AB-AD = DB, AC-AE = EC 7. Partition postulate 8. DB/AD = EC/AE 8. Substitution postulate 9. AD/DB = AE/EC 9. Proportional by Inversion Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally DE || BC AD AE DB EC AD AE DE AB AC BC D E A B C = = = Mr. Chin-Sung Lin

Converse of Triangle Proportionality Theorem
L22_Similar Triangles ERHS Math Geometry Converse of Triangle Proportionality Theorem If the points at which a line intersects two sides of a triangle divide those sides proportionally, then the line is parallel to the third side Given: AD AE DB EC Prove: DE || BC D E A B C = Mr. Chin-Sung Lin

Converse of Triangle Proportionality Theorem
L22_Similar Triangles ERHS Math Geometry Converse of Triangle Proportionality Theorem D E A B C Statements Reasons 1. AD/DB = AE/EC Given 2. DB/AD = EC/AE 2. Proportional by Inversion 3. (DB+AD)/AD = (EC+AE)/AE 3. Proportional by composition 4. DB+AD = AB, EC+AE = AC 4. Partition postulate 5. AB/AD = AC/AE 5. Substitution postulate 6. A  A Reflexive property 7. ABC ~ ADE SAS similarity theorem 8. ABC  ADE 8. Corresponding angles 9. DE || BC Converse of corresponding angles Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem A line is parallel to one side of a triangle and intersects the other two sides if and only if the points of intersection divide the sides proportionally D E A B C Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem
L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, AD = 4, BD = 3, AE = 6, DE = 8 Calculate: CE and BC 8 3 4 6 ? D E A B C Mr. Chin-Sung Lin

Example of Triangle Proportionality Theorem
L22_Similar Triangles ERHS Math Geometry Example of Triangle Proportionality Theorem Given: DE || BC, AD = 4, BD = 3, AE = 6 Calculate: CE and BC 8 3 4 6 4.5 D E A B C 14 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations Mr. Chin-Sung Lin

Dilations A dilation of k is a transformation of the plane such that:
L22_Similar Triangles ERHS Math Geometry Dilations A dilation of k is a transformation of the plane such that: The image of point O, the center of dilation, is O When k is positive and the image of P is P’, then OP and OP’ are the same ray and OP’ = kOP. When k is negative and the image of P is P’, then OP and OP’ are opposite rays and P’ = – kOP Mr. Chin-Sung Lin

Dilations When | k | > 1, the dilation is called an enlargement
L22_Similar Triangles ERHS Math Geometry Dilations When | k | > 1, the dilation is called an enlargement When 0 < | k | < 1, the dilation is called a contraction Under a dilation of k with the center at the origin: P(x, y) → P’(kx, ky) or Dk(x, y) = (kx, ky) Mr. Chin-Sung Lin

Dilations and Similar Triangles
L22_Similar Triangles ERHS Math Geometry Dilations and Similar Triangles For any dilation, the image of a triangle is a similar triangle Y (b,0) Z (c, 0) X (0, a) x y A (0, ka) B (kb, 0) C (kc, 0) Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations and Similar Triangles For any dilation, the image of a triangle is a similar triangle Y (b,0) Z (c, 0) X (0, a) x y A (0, ka) B (kb, 0) C (kc, 0) m XY = – a/b m XZ = – a/c m YZ = 0 m AB = – ka/kb = – a/b m AC = – ka/kc = – a/c m BC = 0 Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations and Similar Triangles For any dilation, the image of a triangle is a similar triangle Y (b,0) Z (c, 0) X (0, a) x y A (0, ka) B (kb, 0) C (kc, 0) XY || AB XZ || AC YZ || BC X  A Y  B Z  C ABC ~ ADE Mr. Chin-Sung Lin

Dilations and Angle Measures
L22_Similar Triangles ERHS Math Geometry Dilations and Angle Measures Under a dilation, angle measure is preserved Y (b,0) Z (c, 0) X (0, a) x y A (0, ka) B (kb, 0) C (kc, 0) Mr. Chin-Sung Lin

Dilations and Angle Measures
L22_Similar Triangles ERHS Math Geometry Dilations and Angle Measures Under a dilation, angle measure is preserved We have proved: Y (b,0) Z (c, 0) X (0, a) x y A (0, ka) B (kb, 0) C (kc, 0) X  A Y  B Z  C Mr. Chin-Sung Lin

Dilations and Midpoint
L22_Similar Triangles ERHS Math Geometry Dilations and Midpoint Under a dilation, midpoint is preserved y y A (0, ka) X (0, a) N M x x Y (b, 0) B (kb, 0) Mr. Chin-Sung Lin

Dilations and Midpoint
L22_Similar Triangles ERHS Math Geometry Dilations and Midpoint Under a dilation, midpoint is preserved y y A (0, ka) X (0, a) N M x x Y (b, 0) B (kb, 0) M(X, Y) = (b/2, a/2) DK (M) = (kb/2, ka/2) N(A, B) = (kb/2, ka/2) N(A, B) = DK (M) Mr. Chin-Sung Lin

Dilations and Collinearity
L22_Similar Triangles ERHS Math Geometry Dilations and Collinearity Under a dilation, collinearity is preserved P (e, f) X (a, b) x y A (ka, kb) Y (c, d) B (kc, kd) Q (ke, kf) Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations and Collinearity Under a dilation, collinearity is preserved P (e, f) X (a, b) x y A (ka, kb) Y (c, d) B (kc, kd) Q (ke, kf) m XP = (b – f) / (a – e) m PY = (f – d) / (e – c) XPY are collinear m XP = m PY (b – f) / (a – e) = (f – d) / (e – c) Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations and Collinearity Under a dilation, collinearity is preserved P (e, f) X (a, b) x y A (ka, kb) Y (c, d) B (kc, kd) Q (ke, kf) m XP = (b – f) / (a – e) m PY = (f – d) / (e – c) XPY are collinear m XP = m PY (b – f) / (a – e) = (f – d) / (e – c) m AQ = (kb – kf) / (ka – ke) = (b – f) / (a – e) m QB = (kf – kd) / (ke – kc) = (f – d) / (e – c) m AQ = m QB AQB are collinear Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations Example The coordinates of parallelogram EFGH are E(0, 0), F(3, 0), G(4, 2), and H(1, 2). Under D3, the image of EFGH is E’F’G’H’. Show that E’F’G’H’ is a parallelogram. Is parallelism preserved? Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Dilations Example The coordinates of parallelogram EFGH are E(0, 0), F(3, 0), G(4, 2), and H(1, 2). Under D3, the image of EFGH is E’F’G’H’. Show that E’F’G’H’ is a parallelogram. Is parallelism preserved? m E’F’ = 0, m G’H’ = 0 m E’F’ = m G’H’ m F’G’ = 2, m H’E’ = 2 m F’G’ = m H’E’ E’F’G’H’ is a parallelogram Parallelism preserved D3 (E) = E’ (0, 0) D3 (F) = F’ (9, 0) D3 (G) = G’ (12, 6) D3 (H) = H’ (3, 6) Mr. Chin-Sung Lin

Proportional Relations among Segments Related to Triangles
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Proportional Relations among Segments Related to Triangles Mr. Chin-Sung Lin

Proportional Relations of Segments
L22_Similar Triangles ERHS Math Geometry Proportional Relations of Segments If two triangles are similar, their corresponding sides altitudes medians, and angle bisectors are proportional Mr. Chin-Sung Lin

Proportional Altitudes
L22_Similar Triangles ERHS Math Geometry Proportional Altitudes If two triangles are similar, the lengths of corresponding altitudes have the same ratio as the lengths of any two corresponding sides A B C X Y Z Mr. Chin-Sung Lin

Proportional Altitudes
L22_Similar Triangles ERHS Math Geometry Proportional Altitudes If two triangles are similar, the lengths of corresponding altitudes have the same ratio as the lengths of any two corresponding sides A B C X Y Z AA Similarity Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Proportional Medians If two triangles are similar, the lengths of corresponding medians have the same ratio as the lengths of any two corresponding sides A B C X Y Z Mr. Chin-Sung Lin

Proportional Medians SAS Similarity
L22_Similar Triangles ERHS Math Geometry Proportional Medians If two triangles are similar, the lengths of corresponding medians have the same ratio as the lengths of any two corresponding sides A B C X Y Z SAS Similarity Mr. Chin-Sung Lin

Proportional Angle Bisectors
L22_Similar Triangles ERHS Math Geometry Proportional Angle Bisectors If two triangles are similar, the lengths of corresponding angle bisectors have the same ratio as the lengths of any two corresponding sides A B C X Y Z Mr. Chin-Sung Lin

Proportional Angle Bisectors
L22_Similar Triangles ERHS Math Geometry Proportional Angle Bisectors If two triangles are similar, the lengths of corresponding angle bisectors have the same ratio as the lengths of any two corresponding sides A B C X Y Z AA Similarity Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Application Problem Two triangles are similar. The sides of the smaller triangle have lengths of 4 m, 6 m, and 8 m. The perimeter of the larger triangle is 63 m. Find the length of the shortest side of the larger triangle Mr. Chin-Sung Lin

L22_Similar Triangles ERHS Math Geometry Application Problem Two triangles are similar. The sides of the smaller triangle have lengths of 4 m, 6 m, and 8 m. The perimeter of the larger triangle is 63 m. Find the length of the shortest side of the larger triangle 4x + 6x + 8x = 63 x = 3.5 4x = 14, 6x = 21, 8x = 28 The length of the shortest side is 14 m Mr. Chin-Sung Lin

Concurrence of the Medians of a Triangle
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Concurrence of the Medians of a Triangle Mr. Chin-Sung Lin

L3_Line Segments in Triangles
ERHS Math Geometry Median of a Triangle A segment from a vertex to the midpoint of the opposite side of a triangle C A B A C B A C B Mr. Chin-Sung Lin

Skill Review 2 - Basic Tools for Proof
ERHS Math Geometry Median of a Triangle If BD is the median of ∆ ABC then, AD  CD A C B D Mr. Chin-Sung Lin

Definition of Centroid
L3_Line Segments in Triangles ERHS Math Geometry Definition of Centroid The three medians meet in the centroid or center of mass (center of gravity) A B Centroid C Mr. Chin-Sung Lin

Theorem about Centroid
L3_Line Segments in Triangles ERHS Math Geometry Theorem about Centroid Any two medians of a triangle intersect in a point that divides each median in the ratio 2 : 1 Given: AE and CD are medians of ABC that intersect at P Prove: AP : EP = CP : DP = 2 : 1 A B Centroid C 2 1 D E P Mr. Chin-Sung Lin

Theorem about Centroid
L22_Similar Triangles ERHS Math Geometry A B C 2 1 D E P Theorem about Centroid Statements Reasons 1. AE and CD are medians 1. Given 2. D is the midpoint of AB 2. Definition of medians E is the midpoint of BC 3. AC || DE Midsegment theorem DE = ½ AC 4. AED  EAC, CDE  DCA 4. Alternate interior angles 5. APC ~ EPD AA similarity theorem 6. AC:DE = 2: Exchange extremes 7. AP:EP = CP:DP = AC:DE 7. Corresp. sides proportional 8. AP:EP = CP:DP = 2:1 8. Transitive property Mr. Chin-Sung Lin

Medians Concurrence Theorem
L3_Line Segments in Triangles ERHS Math Geometry Medians Concurrence Theorem The medians of a triangle are concurrent Given: AM, BN, and CL are medians of ABC Prove: AM, BN, and CL are concurrent A B Centroid C M N L P Mr. Chin-Sung Lin

Medians Concurrence Theorem
L3_Line Segments in Triangles ERHS Math Geometry Medians Concurrence Theorem The medians of a triangle are concurrent Given: AM, BN, and CL are medians of ABC Prove: AM, BN, and CL are concurrent Proof: AM and BN intersect at P AP : MP = 2 : 1 AM and CL intersect at P’ AP’ : MP’ = 2 : 1 P and P’ are on AM, and divide that line segment in the ratio 2 : 1 Therefore, P = P’ and AM, BN, and CL are concurrent A B Centroid C M N L P P’ Mr. Chin-Sung Lin

L3_Line Segments in Triangles
ERHS Math Geometry Centroid The centroid divides each median in a ratio of 2:1. B 2 1 1 Centroid 2 2 1 A C Mr. Chin-Sung Lin

Coordinates of Centroid
L18_Equations of Lines ERHS Math Geometry Coordinates of Centroid The coordinates of the vertices of ΔABC are A(0, 0), B(6, 0), and C(0, 3). (a) Find the coordinates of the centroid P of the triangle. (b) Prove that the centroid divides each median in a ratio of 2:1 Mr. Chin-Sung Lin

L18_Equations of Lines ERHS Math Geometry Coordinates of Centroid The coordinates of the vertices of ΔABC are A(0, 0), B(6, 0), and C(0, 3). (a) Find the coordinates of the centroid P of the triangle. (b) Prove that the centroid divides each median in a ratio of 2:1 Answer (a): P(2, 1) Mr. Chin-Sung Lin

L3_Line Segments in Triangles ERHS Math Geometry Coordinates of Centroid If the coordinates of the vertices of a triangle are: A(x1, y1), B(x2, y2), and C(x3, y3), then the coordinates of the centroid are: x1+x2+x y1+y2+y3 A B Centroid C P P ( , ) Mr. Chin-Sung Lin

Proportions in a Right Triangle
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Proportions in a Right Triangle Mr. Chin-Sung Lin

Projection of a Point on a Line
L18_Equations of Lines ERHS Math Geometry Projection of a Point on a Line The projection of a point on a line is the foot of the perpendicular drawn from that point to the line A Projection P Mr. Chin-Sung Lin

Projection of a Segment on a Line
L18_Equations of Lines ERHS Math Geometry Projection of a Segment on a Line The projection of a segment on a line, when the segment is not perpendicular to the line, is the segment whose endpoints are the projections of the endpoints of the given line segment on the line A Projection B P Q Mr. Chin-Sung Lin

Similar Triangles within a Right Triangle
L24_Right Triangle Altitude Theorem ERHS Math Geometry Similar Triangles within a Right Triangle Mr. Chin-Sung Lin

Identify Similar Triangles
L24_Right Triangle Altitude Theorem ERHS Math Geometry Identify Similar Triangles The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to each other and to the original triangle Given: ABC, mB = 90, BD is an altitude Prove: ABC ~ ADB ~ BDC A B C D Mr. Chin-Sung Lin

Prove Similar Triangles
L24_Right Triangle Altitude Theorem ERHS Math Geometry Prove Similar Triangles BAC  DBC, ACB  BCD, then, similar triangles: ABC ~ BDC BAD  CAB, ABD  ACB, then, similar triangles: ADB ~ ABC BAD  CBD, ABD  BCD, then, similar triangles: ADB ~ BDC So, ABC ~ ADB ~ BDC A B C D Mr. Chin-Sung Lin

Identify Corresponding Sides
L24_Right Triangle Altitude Theorem ERHS Math Geometry Identify Corresponding Sides ABC ~ BDC ABC ~ ADB AB AC BC AB AC BC BD BC DC AD AB BD BC2 = AC  DC AB2 = AC  AD ADB ~ BDC AD AB BD BD BC CD BD2 = AD  DC = = = = A B C D = = Mr. Chin-Sung Lin

Right Triangle Altitude Theorem
L24_Right Triangle Altitude Theorem ERHS Math Geometry Right Triangle Altitude Theorem Mr. Chin-Sung Lin

Right Triangle Altitude Theorem (Part I)
L24_Right Triangle Altitude Theorem ERHS Math Geometry Right Triangle Altitude Theorem (Part I) The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. Given: AD is the altitude Prove: AD2 = BD  DC A B C D Mr. Chin-Sung Lin

Right Triangle Altitude Theorem (Part II)
L24_Right Triangle Altitude Theorem ERHS Math Geometry Right Triangle Altitude Theorem (Part II) If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg. Given: AD is the altitude Prove: AB2 = BC  BD Prove: AC2 = BC  DC A B C D Mr. Chin-Sung Lin

L24_Right Triangle Altitude Theorem
ERHS Math Geometry Application Examples Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 1 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z D A B C 8 2 y x z Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 1 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z x = 4 y = 2√5 z = 4√5 D A B C 8 2 y x z Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 2 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z D A B C x 6 12 y z Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 2 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z x = 18 y = 6√3 z = 12√3 D A B C x 6 12 y z Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 3 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z A z D 6 9 y C B x Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 3 Given: ABC, mB = 90, BD is an altitude Solve: x, y, and z x = 6√3 y = 3√3 z = 3 A z D 6 9 y C B x Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 4 Given: ABC, mB = 90, BD is an altitude Solve: w, x, y, and z A y 25 D w z x C B 20 Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 4 Given: ABC, mB = 90, BD is an altitude Solve: w, x, y, and z w = 15 x = 12 y = 9 z = 16 A y 25 D w z x C B 20 Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 5 Given: ABC, mB = 90 Prove: AB2 + BC2 = AC2 (Pythagorean Theorem) (Hint: Apply Triangle Altitude Theorem) D A B C Mr. Chin-Sung Lin

ERHS Math Geometry Application Example 5 Given: ABC, mB = 90 Prove: AB2 + BC2 = AC2 (Pythagorean Theorem) (Hint: Apply Triangle Altitude Theorem) AB2 = AC  AD BC2 = AC  DC AB2 + BC2 = AC  AD + AC  DC = AC  (AD + DC) = AC  AC = AC2 D A B C Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem
ERHS Math Geometry Pythagorean Theorem Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Theorem If a triangle is a right triangle, then the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides (the legs) Given: ABC, mC = 90 Prove: a2 + b2 = c2 A C B a b c Mr. Chin-Sung Lin

Converse of Pythagorean Theorem
L24_Right Triangle Altitude Theorem ERHS Math Geometry Converse of Pythagorean Theorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle Given: a2 + b2 = c2 Prove: ABC is a right triangle with mC = 90 A C B a b c Mr. Chin-Sung Lin

Converse of Pythagorean Theorem
L22_Similar Triangles ERHS Math Geometry Converse of Pythagorean Theorem A C B a b c R T S a b Statements Reasons 1. a2 + b2 = c Given 2. Draw RST with RT = b, ST = a 2. Create a right triangle m T = 90 3. ST2 + RT2 = RS Pythagorean theorem 4. a2 + b2 = RS2, RS2 = c Substitution postulate 5. RS = c Root postulate 6. ABC  RST SSS postulate 7. C  T CPCTC 8. mC = Substitution postulate 9. ABC is a right triangle 9. Definition of a right triangle

ERHS Math Geometry Pythagorean Theorem A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides ABC, mC = 90 if and only if a2 + b2 = c2 A C B a b c Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example What is the length of the altitude to the base of an isosceles triangle if the length of the base is 24 centimeters and the length of a leg is 15 centimeters? A C B 15 D 24 ? Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example What is the length of the altitude to the base of an isosceles triangle if the length of the base is 24 centimeters and the length of a leg is 15 centimeters? 9 cm A C B 15 D 24 ? Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example If a cone has a height of 24 centimeters and the radius of the base is 10 centimeters, what is the slant height of the cone? A C 24 ? B 10 Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example If a cone has a height of 24 centimeters and the radius of the base is 10 centimeters, what is the slant height of the cone? 26 cm A C 24 ? B 10 Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example If a right prism has a length of 15 cm, a width of 12 cm and a height of 16 cm, what is the length of AB? 12 15 16 A B ? Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Example If a right prism has a length of 15 cm, a width of 12 cm and a height of 16 cm, what is the length of AB? 25 cm 12 15 16 A B ? Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Triples Mr. Chin-Sung Lin

ERHS Math Geometry Pythagorean Triples When three integers can be the lengths of the sides of a right triangle, this set of numbers is called a Pythagorean triple {3, 4, 5} or {3x, 4x, 5x} {5, 12, 13} or {5x, 12x, 13x} {7, 24, 25} or {7x, 24x, 25x} {8, 15, 17} or {8x, 15x, 17x} {9, 40, 41} or {9x, 40x, 41x} Mr. Chin-Sung Lin

Special Right Triangles
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Special Right Triangles Mr. Chin-Sung Lin

45-45-Degree Right Triangle
L24_Right Triangle Altitude Theorem ERHS Math Geometry 45-45-Degree Right Triangle An isosceles right triangle with degree angles A C B 45o √ 2 1 45o 1 Mr. Chin-Sung Lin

30-60-Degree Right Triangle
L24_Right Triangle Altitude Theorem ERHS Math Geometry 30-60-Degree Right Triangle A right triangle with degree angles A 60o 2 1 30o B C √ 3 Mr. Chin-Sung Lin

Special Right Triangle Example
L24_Right Triangle Altitude Theorem ERHS Math Geometry Special Right Triangle Example An isosceles right triangle with a hypotenuse of 4 cm, what is the length of each leg? A C B 4 ? Mr. Chin-Sung Lin

L24_Right Triangle Altitude Theorem ERHS Math Geometry Special Right Triangle Example An isosceles right triangle with a hypotenuse of 4 cm, what is the length of each leg? A C B 4 2√ 2 2√ 2 Mr. Chin-Sung Lin

L24_Right Triangle Altitude Theorem ERHS Math Geometry Special Right Triangle Example A right triangle with a hypotenuse of 4 cm, and one angle of 30o, what is the length of each leg? A 4 ? 30o C B ? Mr. Chin-Sung Lin

L24_Right Triangle Altitude Theorem ERHS Math Geometry Special Right Triangle Example A right triangle with a hypotenuse of 4 cm, and one angle of 30o, what is the length of each leg? A 4 2 30o C B 2√ 3 Mr. Chin-Sung Lin

Triangle Angle Bisector Theorem
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem If an angle bisector divides one side of a triangle into two line segments, then these two line segments and the other two sides are proportional 1  2 AB BD AC DC D 1 A B C 2 = Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem If an angle bisector divides one side of a triangle into two line segments, then these two line segments and the other two sides are proportional 1  2 AB BD AC DC D 1 A B C 2 E 3 = Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry D 1 A B C 2 E 3 Triangle Angle Bisector Theorem Statements Reasons 1. Extend BA and draw EC || AD 1. Create similar triangle 2. 1   Given 3. 2   Alternate interior angles 4. 1  E Corresponding angles 5. E   Transitive property 6. AC = AE Conv. of base angle theorem 7. AB/AE = BD/DC 7. Triangle proporatationality 8. AB/AC = BD/DC Substitution property Mr. Chin-Sung Lin

Triangle Angle Bisector Theorem - Example 1
L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem - Example 1 If 1  2, AB = 6, AC = 4 and BD = 4, DC = ? D 1 A B C 2 6 4 ? Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem - Example 1 If 1  2, AB = 6, AC = 4 and BD = 4, DC = ? AB BD AC DC DC DC = 8/3 = A = 1 2 6 4 C B 4 D 8/3 Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem - Example 2 If 1  2, AB = 8, DC = 10 and BD = 6, AC = ? C 10 ? D 6 1 2 A B 8 Mr. Chin-Sung Lin

L23_Triangle Angle Bisector Theorem ERHS Math Geometry Triangle Angle Bisector Theorem - Example 2 If 1  2, AB = 8, DC = 10 and BD = 6, AC = ? AC DC AB BD AC AC = 40/3 C = 10 = 40/3 D 6 1 2 A B 8 Mr. Chin-Sung Lin

ERHS Math Geometry Application Problems Mr. Chin-Sung Lin

ERHS Math Geometry Application Problem 1 If 1  2, AC = 6, CD = 3, AB = 3x -2, BD = x + 1 Calculate AB D 1 A B C 2 3x - 2 6 x + 1 3 Mr. Chin-Sung Lin

ERHS Math Geometry Application Problem 1 If 1  2, AC = 6, CD = 3, AB = 3x -2, BD = x + 1 Calculate AB x = 4 AB = 10 D 1 A B C 2 3x - 2 6 x + 1 3 Mr. Chin-Sung Lin

ERHS Math Geometry Application Problem 2 If 1  2, AB = 6, AC = 4, BD = x, DC = x - 1 Calculate BC A 1 2 6 4 C B x D x - 1 Mr. Chin-Sung Lin

ERHS Math Geometry Application Problem 2 If 1  2, AB = 6, AC = 4, BD = x, DC = x - 1 Calculate BC x = 3 BC = 5 A 1 2 6 4 C B x D x - 1 Mr. Chin-Sung Lin

Proportions & Products of Line Segments
Similar Triangles ERHS Math Geometry Proportions & Products of Line Segments Mr. Chin-Sung Lin

Key Questions Proving line segments are in proportion
Similar Triangles ERHS Math Geometry Key Questions Proving line segments are in proportion Proving line segments have geometric mean (mean proportional) Proving products of line segments are equal Mr. Chin-Sung Lin

Steps of Proofs Forming a proportion from a product
Similar Triangles ERHS Math Geometry Steps of Proofs Forming a proportion from a product Selecting triangles containing the line segments Identifying and proving the similar triangles (Draw lines if necessary) Proving the line segments are proportional Mr. Chin-Sung Lin

Example - In Proportions
Similar Triangles ERHS Math Geometry Example - In Proportions Given: ABC  DEC Prove: AC BC CD CE = A B C D E Mr. Chin-Sung Lin

Similar Triangles ERHS Math Geometry Example - Geometric Mean Given: ABC  BDC Prove: BC is geometric mean between AC and DC A B C D Mr. Chin-Sung Lin

Example - Equal Product
Similar Triangles ERHS Math Geometry Example - Equal Product Given: AB || DE Prove: AD * CE = BE * DC A D C E B Mr. Chin-Sung Lin

Proportionality Theorem Review
Similar Triangles ERHS Math Geometry Proportionality Theorem Review Mr. Chin-Sung Lin

Triangle Proportionality Theorem (Side-Splitter Theorem)
Similar Triangles ERHS Math Geometry Triangle Proportionality Theorem (Side-Splitter Theorem) If a line parallel to one side of a triangle intersects the other two sides, then it divides them proportionally DE || BC AD AE DB EC AD AE DE AB AC BC D E A B C = = = Mr. Chin-Sung Lin

Similar Triangles ERHS Math Geometry Triangle Angle Bisector Theorem If an angle bisector divides one side of a triangle into two line segments, then these two line segments and the other two sides are proportional 1  2 AB BD AC DC A 1 2 = B C D Mr. Chin-Sung Lin

Similar Triangles Review
ERHS Math Geometry Similar Triangles Review Mr. Chin-Sung Lin

Similar Triangles - AA = = A  X, B  Y ABC~XYZ AB BC CA
ERHS Math Geometry Similar Triangles - AA A  X, B  Y ABC~XYZ AB BC CA XY YZ ZX = = A B C X Y Z Mr. Chin-Sung Lin

Similar Triangles - SAS
ERHS Math Geometry Similar Triangles - SAS A  X, AB/XY = AC/XZ ABC~XYZ AB BC XY YZ = A B C X Y Z Mr. Chin-Sung Lin

Similar Triangles - Parallel Sides & Shared Angle
ERHS Math Geometry Similar Triangles - Parallel Sides & Shared Angle DE || BC ABC~ADE A  A, B  ADE, C  AED AB BC AC AD DE AE = = A B C E D Mr. Chin-Sung Lin

Similar Triangles - Parallel Sides & Vertical Angles
ERHS Math Geometry Similar Triangles - Parallel Sides & Vertical Angles DE || AB ABC~DEC A  D, B  E, ACB  DCE AB BC AC DE EC DC = = E A C D B Mr. Chin-Sung Lin

Similar Triangles - Overlapping Triangles
ERHS Math Geometry Similar Triangles - Overlapping Triangles ABC  BDC ABC~BDC ABC  BDC, A  DBC, C  C AB AC BC BD BC DC = = A B C D Mr. Chin-Sung Lin

Similar Triangles - Angle Bisector
ERHS Math Geometry Similar Triangles - Angle Bisector 1  2, (Draw EC || AD, 2  3) ABD ~ EBC B  B, 1  E, ADB  ECB BA BD AE DC AC DC E = A 1 2 = 3 B C D Mr. Chin-Sung Lin

Application Problems ERHS Math Geometry Mr. Chin-Sung Lin
Similar Triangles ERHS Math Geometry Application Problems Mr. Chin-Sung Lin

Application Problem 1 = Given: BC || DE, 1  2 Prove: EC AE BC AC
Similar Triangles ERHS Math Geometry Application Problem 1 Given: BC || DE, 1  2 Prove: EC AE BC AC = D E A B C 1 2 Mr. Chin-Sung Lin

Application Problem 2 Given: AF and BH are angle bisectors, BC = AC
Similar Triangles ERHS Math Geometry Application Problem 2 Given: AF and BH are angle bisectors, BC = AC Prove: AH * EF = BF * EH F E A B C H Mr. Chin-Sung Lin

ERHS Math Geometry Q & A Mr. Chin-Sung Lin

ERHS Math Geometry The End Mr. Chin-Sung Lin

Ratios, Proportions, and Similarity

Similar presentations

Presentation on theme: "Ratios, Proportions, and Similarity"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Ratios, Proportions, and Similarity

Similar presentations

Presentation on theme: "Ratios, Proportions, and Similarity"— Presentation transcript:

Similar presentations

About project

Feedback