Question

1. solve for the missing lengths in the sets of similar figures below. a. abcd ~ jklm b. △nop ~ △xyz

Explanation:

Step1: Set up proportion for rectangles

For similar rectangles \(ABCD\) and \(JKLM\), the ratios of corresponding sides are equal. So, \(\frac{AB}{JK}=\frac{AD}{JM}\). Substituting the given values, we have \(\frac{12}{x}=\frac{6}{10}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{12}{x}=\frac{6}{10}\) gives us \(6x = 12\times10\).

Step3: Solve for \(x\)

First, calculate \(12\times10 = 120\). Then, \(6x=120\), so \(x=\frac{120}{6}=20\) mm.

For similar triangles \(\triangle NOP\) and \(\triangle XYZ\), assume the side of length \(3\) mm in \(\triangle NOP\) corresponds to the side of length \(39\) mm in \(\triangle XYZ\), and the side of length \(7\) mm in \(\triangle NOP\) corresponds to side \(w\) in \(\triangle XYZ\).

Step4: Set up proportion for triangles

The proportion of corresponding sides is \(\frac{3}{39}=\frac{7}{w}\).

Step5: Cross - multiply

Cross - multiplying gives \(3w=7\times39\).

Step6: Solve for \(w\)

Calculate \(7\times39 = 273\). Then \(3w = 273\), so \(w=\frac{273}{3}=91\) mm.

Answer:

a. \(x = 20\) mm
b. \(w = 91\) mm