Find the unknown lengths in the pair of similar triangles.
A
B
C
R
Q
P
16 in.
40 in.
a
b
12 in.
15 in.
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Part 1
The length of a is 
  
enter your response here in. 

To find the unknown lengths in the pair of similar triangles, we can use the property that corresponding sides of similar triangles are proportional.

Given:

\[
\frac{\text{AB}}{\text{RQ}} = \frac{\text{BC}}{\text{QP}} = \frac{\text{AC}}{\text{RP}}
\]

Let’s assign values:

- \( AB = 16 \, \text{in.}, RQ = 12 \, \text{in.} \)
- \( BC = 40 \, \text{in.}, QP = 15 \, \text{in.} \)
- \( AC = a \), and we need to find \( a \)

### Step 1: Set up the proportion for the corresponding sides
\[
\frac{AB}{RQ} = \frac{AC}{RP}
\]

Substitute the known values:
\[
\frac{16}{12} = \frac{a}{b}
\]

### Step 2: Set up the second proportion to find \( b \)

\[
\frac{BC}{QP} = \frac{AC}{RP}
\]
Substitute the values:
\[
\frac{40}{15} = \frac{a}{b}
\]

### Step 3: Solve for \( a \)

Find the unknown lengths in the pair of similar triangles. A, B, C, R, Q, P, 16 in., 40 in., a, b, 12 in., 15 in. The length of a is.

Learn how to find the unknown lengths of similar triangles using proportionality. This problem involves calculating sides of triangles with given measurements like 16 in., 40 in., 12 in., and 15 in. using key geometric formulas. Suitable for students in Grades 8-10.

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