Question

1. $\triangle mnp \sim \triangle qrp$

(the image shows triangle mnp similar to triangle qrp, with mp = 24, pq = 28, np = x + 8, and rp = 3x - 9)

Explanation:

Step1: Use Similar Triangles Proportion

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

Step2: Cross - Multiply

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

Step3: Expand Both Sides

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

Step4: Subtract \(28x\) from Both Sides

\(72x-28x-216=28x - 28x+224\), which simplifies to \(44x-216 = 224\).

Step5: Add 216 to Both Sides

\(44x-216 + 216=224 + 216\), so \(44x=440\).

Step6: Solve for \(x\)

Divide both sides by 44: \(x=\frac{440}{44}=10\).

Answer:

\(x = 10\)