Question

1. segment ab is parallel to segment ab.

a. what is the length of segment ab?
b. what is the length of segment bb?

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \(A'B'\parallel AB\), we have \(\frac{CA'}{AA'}=\frac{CB'}{BB'}\). Given \(CA' = 6\), \(AA'=3\), and \(CB' = 7\). Let \(BB'=x\). Then \(\frac{6}{3}=\frac{7 + x}{x}\).

Step2: Cross - multiply

Cross - multiplying gives \(6x=3(7 + x)\).

Step3: Expand and solve for \(x\)

Expand: \(6x = 21+3x\). Subtract \(3x\) from both sides: \(6x-3x=21\), so \(3x = 21\), and \(x = 7\).

Step4: For part (a)

Use the ratio of similar triangles. The ratio of similarity \(k=\frac{CA'}{CA}=\frac{6}{6 + 3}=\frac{2}{3}\). If \(A'B' = 5\), and since \(\triangle CA'B'\sim\triangle CAB\), then \(\frac{A'B'}{AB}=\frac{CA'}{CA}\). So \(AB=\frac{3}{2}\times5 = 7.5\).

Answer:

a. The length of segment \(AB\) is \(7.5\).
b. The length of segment \(BB'\) is \(7\).