GEOMETRY CHAPTER 3

Geometry & Measurement 3.1 Measuring Distance, Area and Volume 3.2 Applications and Problem Solving 3.3 Lines, Angles and Triangles

3.1 Rounding Measurements To round: 1. Underline the place 2. If number to the right of the under- lined place is 5 or more, add one 3. Otherwise, do not change 4. Change all digits to the right of underlined number to zeros

3.1 Rounding Example First number to the right of 8 is “6”, so add one to 8 4. Change all digits to the right to 0’s. The answer is, or 39 Example: Round centimeters to the nearest centimeter

3.1 Calculating Distances Linear Measure - a distance which could be around a polygon (perimeter) or around a circle (circumference) Perimeter - sum of the lengths of the sides Circumference - distance around circle

Measure can be in U.S. system (yd, ft, etc.) or metric (cm,m, etc) 3.1 Metric Measures King Milk HenryDied Drinking Monday Chocolate kmhmdamm dmcmmm kilometerhectometerdekameter meter decimeter centimetermillimeter

3.1 Metric Measures 1 cm = 0.01 m 1 dm = 0.1 m 1 mm = m 1 hm = 100 m 1 km = 1000 m 1 dam = 10 m

3.1 Linear Distance 2. What is the distance around the polygon, in meters? 75 cm 78 cm 95 cm 80 cm = 328 cm km A. 328 m hmdammdmcmmm B m C m D m

3.1 Calculating Areas Rectangle Parallelogram Square Triangle Trapezoid Circle r

3.1 Area - Square Units 4. What is the area of a circular region whose diameter is 6 cm? If d = 6, then r = 3 Formula:

Surface area of a rectangular solid 3.1 Examples of Area L W H There are 6 faces of the solid A=2LH Front/back Sides (Left/Right) +2WH Top/Bottom +2LWSquare units

3.1 Examples of Area 6. What is the surface area of a rectangular solid that is 12 in. long, 5 in. wide and 6 in. high? D. 360 sq. in. L=12 W=5 H=6 A=2(12)(6)+2(5)(6)+2(12)(5) A=2LH+2WH +2LW A. 360 cubic in. B. 324 sq. in. C. 324 cubic in.

3.1 Volume - Cubic Units Rectangular Solid Cylinder h h h Cone Sphere r r r w l

3.1 Example of Volume 8. What is the volume of a sphere with a 12 inch diameter? Formula: If d = 12, then r = 6 Since (6)(6)(6)= 216, the only reasonable ans. is C

3.1 Identifying the Unit 9. Which of the following would not be used to measure the amount of water needed to fill a swimming pool? A. Cubic feet Think of “volume” as capacity or filling up the inside of a 3D figure. linear B. Liters C. GallonsD. Meters

3.2 Application Example 1. What will be the cost of tiling a room measuring 12 ft. by 15 ft. if square tiles cost $2 each & measure 12 in.? Since 12 inches = 1 ft, one tile is 1 ft on each side or 1 sq. ft. Area room: A = bh; (12)(15) = 180 sq ft And (180)($2) = $360 cost A. $180 B. $4320C. $360 D. $3600

3.2 Pythagorean Theorem For any RIGHT TRIANGLE c a b Side opposite the right angle is the hypotenuse “c”

3.2 Pythagorean Theorem 3.A TV antenna 12 ft. high is to be anchored by 3 wires each attached to the top of antenna and to pts on the roof 5 ft. from base of the antenna. If wire costs $.75 per ft, what will be the cost? 12 Cost is.75 x 39 =$ c A. $27.00 B. $29.25 C. $9.75 D. $38.25

3.2 Infer & Select Formulas 7. The figure shows a regular hexagon Select the formula for total area b Total area is the area of the 6 identical triangles. A. 3h+b If area of 1 triangle = 1/2xbh, then 6 x 1/2 x bh = 3 bh B. 6(h+b)C. 6hbD. 3hb h

3.3 Lines; Angles; Triangles straight angle 180 right angle 90 obtuse > 90, < 180 acute angle < 90 comp. sum to 90 supp. sum to 180 vertical angles-equal ANGLES TRIANGLES Right triangle Acute triangle Obtuse triangle Scalene triangle Isosceles Equilateral

3.3 Properties Example 2. What type of triangle is ABC? A. Isosceles Since sum of angles of triangle = 180, and = 125, then angle C = = 55. If 2 angles =, then isosceles. C B. Right C. Equilateral D. Scalene

3.3 Angle Measures B B B B S S S S 1. B S Theorem All B’s are =, All S’s are = B + S = Perpendicular lines intersect to form right angles.

3.3 Angle Measures 7 L1L1 1 Terminology The parallel lines are cut by transversal T L2L Corresponding angles are = 1 and 5, 3 and 7, 2 and 6, 4 and 8 Vertical angles are = 1 and 4, 3 and 2, 6 and 7, 5 and 8 T

3.3 Angle Measures 7 L1L1 1 Terminology The parallel lines are cut by transversal T L2L T Alternate interior angles are = 4 and 5, 3 and 6 Alternate exterior angles are = 1 and 8, 2 and 7

3.3 Angle Measures 3. If 2 angles of a triangle are =, then sides opposite are = 4. If 2 sides of a triangle are =, then angles opposite are =

3.3 Examples 75   Which statement is true for the figure shown at the right given that L 1 and L 2 are parallel? After using the BS theorem, angle T does = 75 and angle S=105 T S V R L1L1 L2L

3.3 Similar Triangles Two triangles are similar if all angles are = and sides proportional 10.Which statements are true? i. m  A = m  E ii. AC = 6 iii. CE/CA = CB/CD A. i only B. ii only C. i and ii only D. i, ii, iii A x E 5 C D B 4 Since m  D=m  B and  DCE and  ACB are Vertical angles m  A=m  E

3.3 Similar Triangles Two triangles are similar if all angles are = and sides proportional 10.Which statements are true? i. m  A = m  E ii. AC = 6 iii. CE/CA = CB/CD A. i only B. ii only C. i and ii only D. i, ii, iii A x E 5 C D B 4 The triangle are similar, thus ratios of corresponding sides are =. x/4 = 7.5/5 thus x= 4(7.5)/5 = 6

3.3 Similar Triangles Two triangles are similar if all angles are = and sides proportional 10.Which statements are true? i. m  A = m  E ii. AC = 6 iii. CE/CA = CB/CD A. i only A x E 5 C D B 4 The triangle are similar, thus ratios of corresponding sides are =. CE/CA = CD/CB thus iii is false! B. ii onlyC. i and ii onlyD. i, ii, iii

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