Presentation on theme: "Objective: Find and simplify the ratio of two numbers."— Presentation transcript:
1 Objective: Find and simplify the ratio of two numbers. 7.1 Ratio and ProportionObjective: Find and simplify the ratio of two numbers.
2 Computing RatiosIf a and b are two quantities that are measured in the same units, then the ratio of a to b is a/b. The ratio of a to b can also be written as a:b. Ratios must be SIMPLIFIED. For instance, the ratio of 6:8 is usually simplified to 3:4. (You divided by 2)
3 Ex. 1: Simplifying Ratios Simplify the ratios:12 cm b. 6 ft c. 9 in.4 cm ft in.
4 Ex. 1: Simplifying Ratios Simplify the ratios:12 cm b. 6 ft4 m inSolution: To simplify the ratios with unlike units, convert to like units so that the units divide out. Then simplify the fraction, if possible.
5 Ex. 1: Simplifying Ratios Simplify the ratios:12 cm4 m12 cm 12 cm 12 34 m ∙100cm
6 Ex. 1: Simplifying Ratios Simplify the ratios:b. 6 ft18 in6 ft 6∙12 in 72 in18 in in. 18 in. 1
7 Ex. 3: Using Extended Ratios The measures of the angles in ∆JKL are in the extended ratio 1:2:3. Find the measures of the angles.Begin by sketching a triangle. Then use the extended ratio of 1:2:3 to label the measures of the angles as x°, 2x°, and 3x°.2x°3x°x°
8 Solution: Statement x°+ 2x°+ 3x° = 180° 6x = 180 x = 30 Reason Triangle Sum TheoremCombine like termsDivide each side by 6So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.
9 Using ProportionsAn equation that equates two ratios is called a proportion. For instance, if the ratio of a/b is equal to the ratio c/d; then the following proportion can be written:MeansExtremes = The numbers a and d are the extremes of the proportions. The numbers b and c are the means of the proportion.
10 Properties of proportions CROSS PRODUCT PROPERTY. The product of the extremes equals the product of the means.If = , then ad = bc
11 Properties of proportions RECIPROCAL PROPERTY. If two ratios are equal, then their reciprocals are also equal.If = , then = baTo solve the proportion, you find the value of the variable.
12 Ex. 5: Solving Proportions 45Write the original proportion.Reciprocal prop.Multiply each side by 4Simplify.=x74x74=4528x=5
13 Ex. 5: Solving Proportions 32Write the original proportion.Cross Product prop.Distributive PropertySubtract 2y from each side.=y + 2y3y = 2(y+2)3y = 2y+4y4=
14 Objective: To identify and apply similar polygons.
15 Identifying similar polygons When corresponding angles of two polygons are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.The symbol ~ is used to indicate similarity. So, ABCD ~ EFGH.
17 Ex. 1: Writing Similarity Statements Pentagons JKLMN and STUVW are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.
18 Ex. 1: Writing Similarity Statements Because JKLMN ~ STUVW, you can write J S, K T, L U, M V AND N W.You can write the proportionality statement as follows:JKKLLMMNNJ====STTUUVVWWS
19 Ex. 2: Comparing Similar Polygons Decide whether the figures are similar. If they are similar, write a similarity statement.
20 SOLUTION:As shown, the corresponding angles of WXYZ and PQRS are congruent. Also, the corresponding side lengths are proportional.WX153==PQ102YZ93XY63====RS62QR42WX153==So, the two figures are similar and you can write WXYZ ~ PQRS.PQ102
21 Using similar polygons in real life If two polygons are similar, then the ratio of lengths of two corresponding sides is called the scale factor. In Example 2 on the previous page, the common ratio of is the scale factor of WXYZ to PQRS.32
22 Ex. 4: Using similar polygons The rectangular patio around a pool is similar to the pool as shown. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.16 ft24 ft32 ft48 ft
23 Because the rectangles are similar, the scale factor of the patio to the pool is 48 ft: 32 ft. , which is 3:2 in simplified form.The perimeter of the patio is 2(24) + 2(48) = 144 feet and the perimeter of the pool is 2(16) + 2(32) = 96 feet The ratio of the perimeters is16 ft24 ft32 ft48 ft1443, or962
24 Theorem 8.1: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding parts.If KLMN ~ PQRS, thenKL + LM + MN + NK=PQ + QR + RS + SPKLLMMNNK===PQQRRSSP
25 Ex. 1: Writing Proportionality Statements In the diagram, ∆BTW ~ ∆ETC.Write the statement of proportionality.34°ETTCCE==BTTWWB79°
26 Ex. 1: Writing Proportionality Statements In the diagram, ∆BTW ~ ∆ETC.Find mTEC.B TEC, SO mTEC = 79°34°79°
27 Ex. 1: Writing Proportionality Statements In the diagram, ∆BTW ~ ∆ETC.Find ET and BE.34°CEETWrite proportion.=WBBT3ETSubstitute values.=12203(20)ETMultiply each side by 20.=79°125=ETSimplify.Because BE = BT – ET, BE = 20 – 5 = 15. So, ET is 5 units and BE is 15 units.