Question

practice 1
in the diagram, segment ( ab ) is parallel to segment ( ab ).

a. what is the length of segment ( ab )?
type your answer in the box.

units
b. what is the length of segment ( bb )?
type your answer in the box.

units

Explanation:

Step1: Use the similarity of triangles

Since \(A'B'\parallel AB\), \(\triangle CA'B'\sim\triangle CAB\). The ratio of corresponding sides is equal. Let \(AB = x\) and \(B'B=y\). The ratio of sides is \(\frac{CA'}{CA}=\frac{A'B'}{AB}=\frac{CB'}{CB}\). We know that \(CA' = 6\), \(CA=6 + 3=9\), \(A'B' = 5\), \(CB'=7\), \(CB=7 + y\).

Step2: Find the length of \(AB\)

Using \(\frac{CA'}{CA}=\frac{A'B'}{AB}\), substitute the values: \(\frac{6}{9}=\frac{5}{AB}\). Cross - multiply gives \(6AB=9\times5\). Then \(AB=\frac{9\times5}{6}=\frac{45}{6} = 7.5\).

Step3: Find the length of \(B'B\)

Using \(\frac{CA'}{CA}=\frac{CB'}{CB}\), substitute the values: \(\frac{6}{9}=\frac{7}{7 + y}\). Cross - multiply: \(6(7 + y)=9\times7\). Expand: \(42+6y = 63\). Subtract 42 from both sides: \(6y=63 - 42=21\). Divide by 6: \(y=\frac{21}{6}=3.5\).

Answer:

a. \(7.5\) units
b. \(3.5\) units