Question

a) what is the scale factor, k, from triangle a to triangle b?
b) find the value of x.
c) find the value of y.
triangles: triangle b has two sides labeled 10.4, one side labeled y; triangle a has two sides labeled 13, 15, one side labeled x

Explanation:

Part a)

Step1: Identify corresponding sides

Triangle A has a side of length 13, and Triangle B has a corresponding side of length 10.4. The scale factor \( k \) from Triangle A to Triangle B is the ratio of the length of the side in B to the length in A.
\( k = \frac{10.4}{13} \)

Step2: Calculate the ratio

Simplify \( \frac{10.4}{13} = 0.8 \)

Step1: Use scale factor for side \( x \)

The side corresponding to \( x \) in Triangle B is 10.4. The scale factor from A to B is 0.8, so \( \frac{x}{10.4}=\frac{1}{0.8} \) (since scale factor \( k = \frac{\text{length in B}}{\text{length in A}} \), so \( \text{length in A}=\frac{\text{length in B}}{k} \))

Step2: Solve for \( x \)

\( x=\frac{10.4}{0.8}=13 \)? Wait, no, wait. Wait, Triangle A has sides 13 and 15, Triangle B has two sides 10.4. Wait, maybe I mixed up. Wait, scale factor from A to B is \( k = \frac{10.4}{13}=0.8 \). So for side \( x \) in A, corresponding side in B is 10.4? Wait, no, let's re-express. Let's say Triangle A: sides 13, 15, \( x \); Triangle B: sides 10.4, \( y \), 10.4. So corresponding sides: 13 (A) corresponds to 10.4 (B), 15 (A) corresponds to \( y \) (B), \( x \) (A) corresponds to 10.4 (B). So scale factor \( k = \frac{\text{B side}}{\text{A side}}=\frac{10.4}{13}=0.8 \). So to find \( x \) (A side), since B side is 10.4, \( x=\frac{10.4}{k}=\frac{10.4}{0.8}=13 \)? Wait, no, that can't be. Wait, maybe the corresponding sides: 13 (A) ↔ 10.4 (B), \( x \) (A) ↔ 10.4 (B), 15 (A) ↔ \( y \) (B). So since 13 (A) maps to 10.4 (B), then \( x \) (A) maps to 10.4 (B), so \( x = 13 \)? Wait, no, that would mean two sides of A are 13, so A is isoceles? Wait, maybe I made a mistake. Wait, let's do it again. Scale factor from A to B is \( k=\frac{\text{length in B}}{\text{length in A}} \). So for the side of length 13 in A, the corresponding side in B is 10.4. So \( k = \frac{10.4}{13}=0.8 \). Now, the side \( x \) in A corresponds to the side of length 10.4 in B. So \( \text{length in B}=k\times\text{length in A} \), so \( 10.4 = 0.8\times x \). Then \( x=\frac{10.4}{0.8}=13 \). Wait, that's the same as the other side. So Triangle A has two sides 13 and one side 15, so it's isoceles, and Triangle B has two sides 10.4 and one side \( y \), so it's also isoceles. That makes sense.

Step1: Use scale factor for side \( y \)

The side of length 15 in A corresponds to side \( y \) in B. The scale factor \( k = 0.8 \), so \( y = k\times15 \)

Step2: Calculate \( y \)

\( y = 0.8\times15 = 12 \)

Answer:

\( 0.8 \)

Part b)