Question

1. \\( \triangle mnp \sim \triangle qrp \\); find \\( pr \\)

Explanation:

Step1: Use the property of similar triangles

Since \(\triangle MNP\sim\triangle QRP\), the ratios of corresponding sides are equal. So, \(\frac{NP}{RP}=\frac{MP}{QP}\). Substituting the given values, we get \(\frac{x + 8}{3x-9}=\frac{24}{28}\).

Step2: Cross - multiply

Cross - multiplying gives \(28(x + 8)=24(3x - 9)\).
Expanding both sides: \(28x+224 = 72x-216\).

Step3: Solve for \(x\)

Subtract \(28x\) from both sides: \(224=72x - 28x-216\), i.e., \(224 = 44x-216\).
Add \(216\) to both sides: \(224 + 216=44x\), so \(440 = 44x\).
Divide both sides by \(44\): \(x = 10\).

Step4: Find \(PR\)

Substitute \(x = 10\) into \(3x-9\). So, \(PR=3\times10 - 9=30 - 9 = 21\).

Answer:

\(21\)