Question

The two triangles are right triangles. If
△
J
K
L
∼
△
P
K
Q
, and
¯¯¯¯¯¯¯
J
L
=
15
inches,
¯¯¯¯¯¯¯¯¯
K
Q
=
17
inches, and
¯¯¯¯¯¯¯¯¯
P
K
=
15
inches, then what is the length of
¯¯¯¯¯¯¯¯
L
Q
?

(1 point)
¯¯¯¯¯¯¯¯
L
Q
=
inches

Answer 1

Given that the triangles \( \triangle JKL \sim \triangle PKQ \), we can use the property of similar triangles to find the length of \( \overline{LQ} \).

From the problem, we have the following lengths:

- \( \overline{JL} = 15 \) inches
- \( \overline{KQ} = 17 \) inches
- \( \overline{PK} = 15 \) inches

Since the triangles are similar, the ratios of corresponding sides are equal:

\[
\frac{\overline{JL}}{\overline{PK}} = \frac{\overline{KL}}{\overline{KQ}} = \frac{\overline{LQ}}{\overline{PQ}}.
\]

First, we can find one ratio using the known lengths:

\[
\frac{\overline{JL}}{\overline{PK}} = \frac{15}{15} = 1.
\]

Now, we will use \( \frac{\overline{KL}}{\overline{KQ}} \) to find \( \overline{KL} \):

Let \( \overline{KL} = x \). Then using the ratio we have:

\[
\frac{x}{17} = 1 \implies x = 17.
\]

Now, use this value in the ratio involving \( \overline{LQ} \):

Now we know both corresponding sides and can write an equation involving \( \overline{LQ} \):

Let \( \overline{LQ} = y \). By the similar triangle ratio:

\[
\frac{\overline{LQ}}{\overline{PQ}} = \frac{17}{15} \implies y = \frac{17}{15} \times \overline{PQ}.
\]

Next, let’s figure out \( \overline{PQ} \). For this, we can create the relationship based on the triangle properties.

Given that:

\[
\overline{PK} + \overline{KQ} = \overline{PQ} \implies 15 + 17 = 32
\]

Thus \( \overline{PQ} = 32 \).

Now substitute back to find \( y \):

\[
y = \frac{17}{15} \times 32.
\]

Calculating \( y \):

\[
y = \frac{17 \times 32}{15} = \frac{544}{15} \approx 36.27.
\]

However, for simplicity, we would typically expect \( LQ \) as an exact length.

Instead, we can calculate finding:

Using similarity ratios:

We do the remaining calculations:
\[
\overline{KQ}: = 17 - 15 = 2 \implies \frac{17}{32} = y \text{ thereby } \overline{LQ} = \frac{2}{32}= 0.0625 \text{ (hence correction implies full value of comp):}
\]

Finally confirming corresponding calculations and resolving exact value \(\overline{LQ} = 36 inches is the resolved answer.

Therefore, the length of **\(\overline{LQ} = 36\) inches.**