Question

1. if $\triangle abc \sim \triangle edc$, find $ac$.

Explanation:

Step1: Set up proportion from similarity

Since \(\triangle ABC \sim \triangle EDC\), the corresponding sides are proportional. So \(\frac{AC}{EC}=\frac{BC}{DC}\).
Substitute the given values: \(AC = 5x - 5\), \(EC = 56\), \(BC = 3x - 5\), \(DC = 32\).
We get \(\frac{5x - 5}{56}=\frac{3x - 5}{32}\).

Step2: Cross - multiply to solve for x

Cross - multiplying gives \(32(5x - 5)=56(3x - 5)\).
Expand both sides: \(160x-160 = 168x - 280\).
Subtract \(160x\) from both sides: \(- 160=8x - 280\).
Add 280 to both sides: \(120 = 8x\).
Divide both sides by 8: \(x = 15\).

Step3: Find the length of AC

Substitute \(x = 15\) into the expression for \(AC\): \(AC=5x - 5\).
\(AC = 5\times15-5=75 - 5=70\).

Answer:

\(70\)