Question

In the figure below, triangle RPQ is similar to triangle RTS.

Triangle R P Q. Side P R is 42 and side P Q is x. Triangle S R T. Side S T is 36 and side R T is 28.

What is the distance between P and Q?
24
42
50
54

Answer 1

Since triangle RPQ is similar to triangle RTS, we can set up a proportion based on the corresponding sides of the triangles.

Given:

- In triangle RPQ:
- Side PR = 42
- Side PQ = x

- In triangle RTS:
- Side ST = 36
- Side RT = 28

Using the properties of similar triangles, we have:

\[
\frac{PQ}{ST} = \frac{PR}{RT}
\]

Substituting the known values into the equation:

\[
\frac{x}{36} = \frac{42}{28}
\]

Now, simplify the right side:

\[
\frac{42}{28} = \frac{3}{2}
\]

So we rewrite the proportion:

\[
\frac{x}{36} = \frac{3}{2}
\]

We can cross-multiply to solve for \( x \):

\[
2x = 3 \cdot 36
\]

Calculating the right side:

\[
2x = 108
\]

Now, divide both sides by 2:

\[
x = \frac{108}{2} = 54
\]

Thus, the distance between P and Q is

\(\boxed{54}\).