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Honors Geometry Unit 5 Similarity and Dilations

Lesson OneProportion and Similarity

ObjectivesI can define similar figures, proportion, ratio

I can find a scale factor between similar figuresWhat is a Ratio?We discussed ratios in Unit 1

To compare two quantities: a and bWe write a : b

Which implies

Ratios do not include units of measurement

RatiosRatios can also be expressed as decimals

In this case, the ratio is referred to as a unit ratio

Ex: Batting Averagehits vs. at bats:

Write and Simplify RatiosSCHOOL The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth.To find this ratio, divide the number of athletes by the total number of students.Answer: The athlete-to-student ratio is 0.3.

0.3 can be written as

ProportionWhen two ratios are set equal to each other, the equation is called a proportion

We solve these equations by cross multiplying

Use Cross Products to Solve ProportionsAnswer: x = 2

Original proportionCross ProductsSimplify.Add 30 to each side.

Divide each side by 24.ABCD

A.n = 9B.n = 8.9C.n = 3D.n = 1.8

Use Proportions to Make PredictionsPETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Moniques school, predict the total number of students with a dog or a cat.Write and solve a proportion that compares the number of students who have a pet to the number of students in the school.

18 ( 870)= 30xCross Products Property15,660= 30xSimplify.522= xDivide each side by 30.Answer: Based on Monique's survey, about 522 students at her school have a dog or a cat for a pet.Why?Multiple figures that have the same shape but are different sizes are known as similar figures

Similar figures have corresponding angles that are congruent

Similar figures have corresponding side lengths that are proportionalSimilar - SymbolTo show that two figures are similar, we use the symbol ~

We will write similarity statements

Use this symbol just as you would = or

ExampleSimilar Polygons

The ratio is the same for all 4 sets of corresponding sidesUse a Similarity StatementIf ABC ~ RST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

ABC ~ RSTCongruent Angles: A R, B S, C T

Use Similar Figures to Find Missing Measures

The two polygons are similar. Find the values of x and y.Use the congruent angles to write the corresponding vertices in order.polygon ABCDE ~ polygon RSTUVAnswer: x = __92y = __313ABCD

A.a = 1.4B.a = 3.75C.a = 2.4D.a = 2The two polygons are similar. Solve for a and b

b = 1.2b = 2.1b = 7.2b = 9.3Scale FactorWhen two figures are similar, the ratio that is found between all sets of side lengths is called the scale factorTypically represented with the letter k

Depends on the order of comparison

If 0 < k < 1, then the scale factor causes the figure to shrink, or reduce in size

If k > 1, then the scale factor causes the figure to grow in size, or enlarge

What happens if k = 1?Scale Factor

Use a Scale Factor to Find PerimeterIf ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon.

Use a Scale Factor to Find PerimeterThe scale factor ABCDE to RSTUV is or . ___AEVU__47Write a proportion to find the length of DC.Since DC AB and AE DE, the perimeter of ABCDE is 6 + 6 + 6 + 4 + 4 or 26.

Write a proportion.4(10.5)= 7 DCCross Products Property6= DCDivide each side by 7.Use a Scale Factor to Find PerimeterUse the perimeter of ABCDE and scale factor to write a proportion. Let x represent the perimeter of RSTUV.

Theorem 7.1Substitution4x= (26)(7)Cross Products Propertyx= 45.5Solve.ABCDA.LMNOP = 40, VWXYZ = 30B.LMNOP = 32, VWXYZ = 24C.LMNOP = 45, VWXYZ = 40D.LMNOP = 60, VWXYZ = 45If LMNOP ~ VWXYZ, find the perimeter of each polygon.