Question

△abc ~ △def. find the values of x and y.

Explanation:

Step1: Set up proportion for side lengths

Since \(\triangle ABC\sim\triangle DEF\), the ratios of corresponding sides are equal.
For sides \(AC\) and \(DF\), \(BC\) and \(EF\), \(AB\) and \(DE\).
\(\frac{AC}{DF}=\frac{BC}{EF}=\frac{AB}{DE}\)
\(\frac{54}{2y + 34}=\frac{45}{20}=\frac{x-4}{16}\)

Step2: Solve for \(y\)

First, use \(\frac{54}{2y + 34}=\frac{45}{20}\)
Cross - multiply: \(54\times20=45\times(2y + 34)\)
\(1080 = 90y+1530\)
Subtract \(1530\) from both sides: \(1080-1530=90y\)
\(- 450=90y\)
Divide both sides by \(90\): \(y=- 5\)

Step3: Solve for \(x\)

Use \(\frac{x - 4}{16}=\frac{45}{20}\)
Cross - multiply: \(20\times(x - 4)=45\times16\)
\(20x-80 = 720\)
Add \(80\) to both sides: \(20x=720 + 80\)
\(20x=800\)
Divide both sides by \(20\): \(x = 40\)

Answer:

\(x = 40,y=-5\)