Question

△abc ~ △def. find the values of x and y.
diagram: △abc (a bottom-left, b bottom-right, c top) with ac=49, ab=x−4, bc=42; △def (d bottom-left, e bottom-right, f top) with de=16, ef=24, df=3y+25. corresponding angles (a&d, b&e, c&f) marked equal.
a) x
b) y

Explanation:

Step1: Identify corresponding sides

Since $\triangle ABC \sim \triangle DEF$, their corresponding sides are proportional. Let's match the sides:

• $AB$ corresponds to $DE$: $AB = x - 4$, $DE = 16$
• $BC$ corresponds to $EF$: $BC = 42$, $EF = 24$
• $AC$ corresponds to $DF$: $AC = 49$, $DF = 3y + 25$

First, find the scale factor using $BC$ and $EF$. The scale factor from $\triangle DEF$ to $\triangle ABC$ is $\frac{BC}{EF} = \frac{42}{24} = \frac{7}{4}$.

Step2: Solve for \( x \)

Using the proportion for $AB$ and $DE$:
\[
\frac{AB}{DE} = \frac{7}{4} \implies \frac{x - 4}{16} = \frac{7}{4}
\]
Cross - multiply:
\[
4(x - 4)=16\times7
\]
\[
4x-16 = 112
\]
Add 16 to both sides:
\[
4x=112 + 16
\]
\[
4x=128
\]
Divide by 4:
\[
x=\frac{128}{4}=32
\]

Step3: Solve for \( y \)

Using the proportion for $AC$ and $DF$:
\[
\frac{AC}{DF}=\frac{7}{4}\implies\frac{49}{3y + 25}=\frac{7}{4}
\]
Cross - multiply:
\[
7(3y + 25)=49\times4
\]
\[
21y+175 = 196
\]
Subtract 175 from both sides:
\[
21y=196 - 175
\]
\[
21y = 21
\]
Divide by 21:
\[
y = 1
\]

Answer:

a) \( x = 32 \)
b) \( y = 1 \)