Question

sp#10 6. if \\( \triangle pqr\sim\triangle utv \\), find the value of \\( x \\).

Explanation:

Step1: Use the property of similar triangles

For similar triangles \(\triangle PQR\sim\triangle UTV\), the ratios of corresponding sides are equal. That is \(\frac{QR}{TV}=\frac{PR}{UV}\). Given \(QR = 8\), \(PR=14\), \(TV=x - 3\), and \(UV=x + 6\).
So we have the proportion \(\frac{8}{x - 3}=\frac{14}{x + 6}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{8}{x - 3}=\frac{14}{x + 6}\) gives us \(8(x + 6)=14(x - 3)\).
Expand both sides: \(8x+48 = 14x-42\).

Step3: Solve for \(x\)

Subtract \(8x\) from both sides: \(48=14x - 8x-42\), which simplifies to \(48 = 6x-42\).
Add \(42\) to both sides: \(48 + 42=6x\), so \(90 = 6x\).
Divide both sides by \(6\): \(x=\frac{90}{6}=15\).

Answer:

\(x = 15\)