https://file.aiquickdraw.com/mathbot/file/1725479937791fxxx8kh0.jpg

To solve for the unknown lengths \(a\) and \(b\) in the pair of similar triangles, we can use the property of similar triangles that states corresponding sides are proportional.

### Given:
- Triangles \(ABC\) and \(PQR\) are similar.
- Side \(AB = 20 \, \text{in}\), \(BC = 36 \, \text{in}\), and \(AC = a\) in triangle \(ABC\).
- Side \(PQ = b\), \(QR = 15 \, \text{in}\), and \(PR = 33 \, \text{in}\) in triangle \(PQR\).

### Step 1: Set up the proportion
Since the triangles are similar, the ratios of corresponding sides are equal:

\[
\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}
\]

### Step 2: Substitute the known values
\[
\frac{20}{b} = \frac{36}{15} = \frac{a}{33}
\]

### Step 3: Solve for \(b\) using \(\frac{20}{b} = \frac{36}{15}\)
\[
\frac{20}{b} = \frac{36}{15}
\]
Cross-multiply to solve for \(b\):
\[
20 \times 15 = 36 \times b
\]
\[
300 = 36b
\]
\[
b = \frac{300}{36} = 8.33 \, \text{in} \, (\text{approximately})
\]

### Step 4: Solve for \(a\) using \(\frac{a}{33} = \frac{36}{15}\)
\[
\frac{a}{33} = \frac{36}{15}
\]
Cross-multiply to solve for \(a\):
\[
a \times 15 = 36 \times 33
\]
\[
15a = 1188
\]
\[
a = \frac{1188}{15} = 79.2 \, \text{in}
\]

### Final Answers:
- \( a = 79.2 \, \text{in} \)
- \( b = 8.33 \, \text{in} \)

Would you like more details or have any questions?

### Related Questions:
1. What other properties of similar triangles can be used in solving problems?
2. How do you determine if two triangles are similar based on their angles?
3. How can the concept of similarity be applied in real-world scenarios?
4. What is the significance of corresponding angles in similar triangles?
5. How do you solve for the height of a triangle if given two similar triangles?

### Tip:
When dealing with similar triangles, always match the corresponding sides and angles carefully to set up the correct proportions.

Find the unknown lengths in the pair of similar triangles.

Solve the problem of finding unknown lengths in a pair of similar triangles by setting up proportions and using the properties of similar triangles. Detailed step-by-step explanation included. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation