Question

1. \\( \triangle m n p \sim \triangle q r p \\); find \\( p r \\)
2. \\( \triangle a b c d \sim \triangle f g e \\); find \\( f e \\)
3. \\( k l m n \sim p q r s \\); find \\( x \\)
4. \\( \triangle r s t \sim \triangle y s z \\); find \\( y z \\)

Explanation:

Step1: Use the property of similar triangles

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

Step2: Cross - multiply

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

Step3: Expand both sides

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

Step4: Move the terms with \(x\) to one side and constants to the other side

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

Step5: Solve for \(x\)

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

Step6: Find \(PR\)

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

Answer:

\(21\)