Question

1. solve for the missing lengths. show all work.

a. △ghi ~ △pqr
b. △abc ~ △xyz

Explanation:

Step1: Set up proportion for similar - triangles

For similar triangles \(\triangle GHI\sim\triangle PQR\), the ratios of corresponding sides are equal. Let's assume the side of length \(7\) in \(\triangle GHI\) corresponds to the side of length \(112\) in \(\triangle PQR\), and the side of length \(2\) in \(\triangle GHI\) corresponds to the side of length \(n\) in \(\triangle PQR\). The proportion is \(\frac{7}{112}=\frac{2}{n}\).

Step2: Cross - multiply

Cross - multiplying the proportion \(\frac{7}{112}=\frac{2}{n}\) gives us \(7n = 2\times112\).

Step3: Solve for \(n\)

First, calculate \(2\times112 = 224\). Then, \(n=\frac{224}{7}=32\).

For part b, assume the side of length \(7\) in \(\triangle ABC\) corresponds to the side of length \(m\) in \(\triangle XYZ\), and the side of length \(23\) in \(\triangle ABC\) corresponds to the side of length \(49\) in \(\triangle XYZ\). The proportion is \(\frac{7}{m}=\frac{23}{49}\).

Step4: Cross - multiply for part b

Cross - multiplying gives \(23m=7\times49\).

Step5: Solve for \(m\)

First, calculate \(7\times49 = 343\). Then, \(m=\frac{343}{23}\approx14.91\).

Answer:

a. \(n = 32\)
b. \(m=\frac{343}{23}\approx14.91\)