Exercise 38 Page 425

We are told that the polygons illustrated on the diagram are similar and asked to find the values of x and y. Let's name the vertices of the polygons for future reference.

First, we need to calculate the scale factor from the bigger polygon to the smaller one. Then we will use it to find the values of x and y.

Scale Factor

Let's recall that a scale factor is the ratio of the lengths of the corresponding sides of the polygons. From the diagram we know that ML corresponds to SR. Calculating the ratio of their lengths, we can find the scale factor from the bigger polygon to the smaller one.

ML/SR=6/4= 3/2

Value of x

Side MN corresponds to SP and the ratio of their lengths is equal to the scale factor. MN/SP= 3/2 Let's substitute MN with x and SP with 5 to solve the equation for x.

MN/SP=3/2

SubstituteII

MN= x, SP= 5

x/5=3/2

MultEqn

LHS * 5=RHS* 5

x=15/2

CalcQuot

Calculate quotient

x=7.5

We got that the value of x is 7.5.

Value of y

To determine the value of y we need the find the measure of ∠ M. Let's recall that in similar polygons corresponding angles are congruent. In our case, ∠ M corresponds to ∠ S in TRSP. We can find the measure of ∠ S using the fact that in every quadrilateral the sum of the interior angles is 360^(∘). m∠ T+ m∠ R+m∠ S+ m∠ P=360^(∘) ⇓ 61^(∘) + 90^(∘) + m∠ S + 116^(∘)=360^(∘) Let's solve this equation for m∠ S.

61^(∘) + 90^(∘) + m∠ S + 116^(∘)=360^(∘)

AddTerms

Add terms

m∠ S + 267^(∘)=360^(∘)

SubEqn

LHS-267^(∘)=RHS-267^(∘)

m∠ S=93^(∘)

We got that ∠ S measures 93^(∘) and, since ∠ M and ∠ S are congruent, so does ∠ M. m∠ M=93^(∘) The measure of ∠ M is represented by the expression (y-73)^(∘). Hence, we can set it equal to 93^(∘) and find the value of y. y-73=93 ⇓ y=166 Therefore, the value of y is 166.