Question

1. $\triangle mnp \sim \triangle qrp$; find $pr$

4.

Explanation:

Step1: Use Similar Triangles Proportion

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

Step2: Cross - Multiply

Cross - multiplying gives \(24(3x - 9)=28(x + 8)\).

Step3: Expand Both Sides

Expanding: \(72x-216 = 28x + 224\).

Step4: Solve for x

Subtract \(28x\) from both sides: \(72x-28x-216=224\), which simplifies to \(44x-216 = 224\). Then add 216 to both sides: \(44x=224 + 216=440\). Divide both sides by 44: \(x = 10\).

Step5: Find PR

Now, \(PR=3x - 9\). Substitute \(x = 10\) into the expression: \(PR=3(10)-9=30 - 9 = 21\).

Answer:

\(21\)