Question

1. if δpqr ~ δutv, find the value of x.

Explanation:

Step1: Set up proportion

Since \(\triangle PQR\sim\triangle UTV\), the ratios of corresponding sides are equal. So, \(\frac{PR}{TV}=\frac{QR}{UV}\). Substituting the given values: \(\frac{14}{x - 3}=\frac{8}{x+6}\).

Step2: Cross - multiply

Cross - multiplying gives \(14(x + 6)=8(x - 3)\).
Expanding both sides: \(14x+84 = 8x-24\).

Step3: Solve for \(x\)

Subtract \(8x\) from both sides: \(14x-8x+84=8x - 8x-24\), so \(6x+84=-24\).
Subtract \(84\) from both sides: \(6x+84 - 84=-24 - 84\), so \(6x=-108\).
Divide both sides by \(6\): \(x=\frac{-108}{6}=-18\). But since length cannot be negative in this context (assuming \(x-3>0\) and \(x + 6>0\) for side lengths), there is a mistake. Let's re - check the proportion.

Wait, actually, the correct proportion should be \(\frac{PR}{TU}=\frac{QR}{VU}\). So \(\frac{14}{x+6}=\frac{8}{x - 3}\).
Cross - multiplying: \(14(x - 3)=8(x + 6)\).
Expanding: \(14x-42=8x + 48\).
Subtract \(8x\) from both sides: \(14x-8x-42=8x-8x + 48\), so \(6x-42=48\).
Add \(42\) to both sides: \(6x-42 + 42=48+42\), so \(6x=90\).
Divide by \(6\): \(x = 15\).

Answer:

\(x = 15\)