Question

1. given the triangles are similar, solve for x.

find the value of x. provide just the number..

Explanation:

Step1: Set up the proportion

Since the triangles are similar, the ratios of corresponding sides are equal. So, \(\frac{JK}{QN}=\frac{HK}{PN}\). Substituting the given values: \(\frac{25}{12}=\frac{6x - 2}{8}\).

Step2: Cross - multiply

Cross - multiplying gives \(25\times8=12\times(6x - 2)\).

$$200 = 72x-24$$

Step3: Solve for \(x\)

Add \(24\) to both sides: \(200 + 24=72x\), so \(224 = 72x\). Then \(x=\frac{224}{72}=\frac{28}{9}\approx3.11\) (This is wrong. Let's re - check the proportion. The correct proportion should be based on the order of similarity. If \(\triangle JHK\sim\triangle QPN\) (by AA similarity as two angles are equal), then \(\frac{JK}{QN}=\frac{HK}{PN}\) is wrong. The correct proportion is \(\frac{JK}{QN}=\frac{HK}{QP}\))

Let's start over:

Step1: Correct proportion

Since \(\triangle JHK\sim\triangle QPN\) (two angles are equal, so AA similarity), \(\frac{JK}{QN}=\frac{HK}{QP}\). Substituting \(JK = 25\), \(QN=12\), \(HK = 6x - 2\), \(QP = 20\). We get \(\frac{25}{12}=\frac{6x - 2}{20}\) is wrong. Wait, no, if \(\triangle JHK\sim\triangle QPN\), then \(\frac{JK}{QN}=\frac{HK}{PN}\) is wrong. Let's use the side - angle - side similarity. The ratio of sides adjacent to the equal angles.

The correct proportion: \(\frac{JK}{QP}=\frac{HK}{PN}\) (because \(\angle J=\angle N\) and \(\angle H=\angle Q\)). So \(\frac{25}{20}=\frac{6x - 2}{8}\)

Step2: Cross - multiply

Cross - multiply: \(25\times8=20\times(6x - 2)\)

$$200=120x-40$$

Step3: Solve for \(x\)

Add \(40\) to both sides: \(200 + 40=120x\), so \(240=120x\). Then \(x = 2\)

Answer:

\(2\)