Question

SIMILAR FIGURES USE THE SIMILARITY RELATIONSHIP TO FIND THE INDICATED VALUE.

Answer 1

We are given two similar triangles, that means that their corresponding sides are at the same proportion with each other, that is:

$\frac{ST}{SZ}=\frac{RT}{YZ}$

Where:

$\begin{gathered} ST=40 \\ SZ=35 \\ RT=3x-7 \\ YZ=2x+2 \end{gathered}$

Replacing the known values, we get:

$\frac{40}{35}=\frac{3x-7}{2x+2}$

Now we will solve for "x". First by simplifying the fraction on the left:

$\frac{8}{7}=\frac{3x-7}{2x+2}$

Now we multiply both sides by (2x+2):

$\frac{8(2x+2)}{7}=3x-7$

Now we multiply both sides by 7:

$8(2x+2)=7(3x-7)$

Now we solve the parenthesis:

$16x+16=21x-49$

Now we subtract 21x on both sides:

$16x-21x+16=-49$

Now we subtract 16 on both sides:

$16x-21x=-49-16$

Solving the operations:

$-5x=-65$

Now we divide both sides by -5

$x=-\frac{65}{-5}=13$

Therefore, the value of x is 13. Now we replace in the equations for the segments, like this:

$\begin{gathered} RT=3x-7 \\ RT=3(13)-7 \\ RT=32 \end{gathered}$

For the other segment:

$\begin{gathered} YZ=2x+2 \\ YZ=2(13)+2 \\ YZ=28 \end{gathered}$