Question

1. klmn ~ pqrs; find x

Explanation:

Step1: Use the property of similar polygons

Since \( KLMN\sim PQRS \), the ratios of corresponding sides are equal. Here, \( LM = 4x + 4 \), \( KN=7x - 9 \), \( SP = 60 \), and assume the ratio of corresponding sides is based on the equality of non - parallel sides (for trapezoids, if they are similar, the ratios of all corresponding sides are equal). So we have the equation \(\frac{4x + 4}{7x - 9}=\frac{48}{48}\) (wait, no, actually, if we consider the non - parallel sides: \(LM\) corresponds to \(QR\) (but no, wait, for similar polygons \(KLMN\sim PQRS\), the side \(LM\) corresponds to \(QR\) (length \(48\)) and \(KN\) corresponds to \(SP\) (length \(60\)). So the proportion is \(\frac{4x + 4}{48}=\frac{7x - 9}{60}\).
Cross - multiply: \(60(4x + 4)=48(7x - 9)\).

Step2: Expand the equation

Expand both sides:

$$
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$$

Step3: Move the terms with \(x\) to one side and constants to the other side

Subtract \(240x\) from both sides: \(240 = 336x-240x - 432\).
\(240=96x - 432\).
Add \(432\) to both sides: \(240 + 432=96x\).
\(672 = 96x\).

Step4: Solve for \(x\)

Divide both sides by \(96\): \(x=\frac{672}{96}\).

Answer:

\(x = 7\)