Exercise 40 Page 590

Apply the Corresponding Angles Theorem, then write a proportion using the Triangle Proportionality Theorem.

x=6, y=4

Practice makes perfect

Let's analyze the given figure. We will start by finding x.

We can see that two segments are congruent, so 4x+2=6x-10. Let's solve this equation to find x.

4x+2=6x-10

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Solve for x

AddEqn

LHS+10=RHS+10

4x+12=6x

SubEqn

LHS-4x=RHS-4x

12=2x

DivEqn

.LHS /2.=.RHS /2.

6=x

RearrangeEqn

Rearrange equation

x=6

Since the two marked right angles are congruent according to the Corresponding Angles Theorem, the segment in the middle of this triangle is parallel to one of its sides.

This means we can use the Triangle Proportionality Theorem.

If a segment parallel to one of the sides of a triangle is drawn between the other sides, the segment divides the other two sides proportionally.

Let's write a proportion using the expressions for the lengths of the segments. 6x-10/4x+2 = 7y-11/3y+5 Since we know that the two segments on the left side are congruent, we know that the two segments on the right side are also congruent. Let's solve the equation 3y+5=7y-11 to find y.

3y+5=7y-11

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Solve for y

AddEqn

LHS+11=RHS+11

3y+16=7y

SubEqn

LHS-3y=RHS-3y

16=4y

DivEqn

.LHS /4.=.RHS /4.

4=y