To find the length of [tex]\( A'B' \)[/tex] when given that triangles [tex]\( ABC \)[/tex] and [tex]\( A'B'C' \)[/tex] are similar, and knowing the lengths of sides [tex]\( AB = 8 \)[/tex] feet, [tex]\( BC = 12 \)[/tex] feet, and [tex]\( B'C' = 13 \)[/tex] feet, follow these steps:
1. Understand the concept of similar triangles:
- When two triangles are similar, their corresponding sides are proportional. This means that the ratios of the lengths of corresponding sides are equal.
2. Identify the given sides and their corresponding counterparts:
- [tex]\( AB \)[/tex] corresponds to [tex]\( A'B' \)[/tex]
- [tex]\( BC \)[/tex] corresponds to [tex]\( B'C' \)[/tex]
3. Set up the proportion based on the corresponding sides:
- Since [tex]\( \frac{AB}{A'B'} = \frac{BC}{B'C'} \)[/tex], we can write:
[tex]\[
\frac{8}{A'B'} = \frac{12}{13}
\][/tex]
4. Solve for [tex]\( A'B' \)[/tex]:
- To find [tex]\( A'B' \)[/tex], cross-multiply to get:
[tex]\[
8 \times 13 = 12 \times A'B'
\][/tex]
- Simplify the left side:
[tex]\[
104 = 12 \times A'B'
\][/tex]
- Divide both sides by 12 to isolate [tex]\( A'B' \)[/tex]:
[tex]\[
A'B' = \frac{104}{12}
\][/tex]
- Perform the division:
[tex]\[
A'B' = 8.666666666666666
\][/tex]
Therefore, the length of [tex]\( A'B' \)[/tex] is approximately [tex]\( 8.67 \)[/tex] feet.
