To determine the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2, we need to use the properties of similar triangles.
1. Identify the known lengths of the triangles:
- Triangle 1:
- Distance along the ground: [tex]\(24\)[/tex] feet
- Distance up the building: [tex]\(32\)[/tex] feet
- Triangle 2:
- Distance along the ground: [tex]\(9\)[/tex] feet
- Distance up the building: unknown
2. Set up the ratio using similar triangles:
Since the triangles are similar, the ratios of the corresponding sides will be equal.
[tex]\[
\frac{\text{Vertical length of Triangle 1}}{\text{Ground length of Triangle 1}} = \frac{\text{Vertical length of Triangle 2}}{\text{Ground length of Triangle 2}}
\][/tex]
Substitute the known lengths:
[tex]\[
\frac{32}{24} = \frac{\text{Vertical length of Triangle 2}}{9}
\][/tex]
3. Solve for the unknown vertical length for Triangle 2:
[tex]\[
\frac{32}{24} = \frac{\text{Vertical length of Triangle 2}}{9}
\][/tex]
To isolate the vertical length of Triangle 2, we multiply both sides by [tex]\(9\)[/tex]:
[tex]\[
\frac{32}{24} \times 9 = \text{Vertical length of Triangle 2}
\][/tex]
4. Perform the multiplication:
[tex]\[
\text{Vertical length of Triangle 2} = \frac{32 \times 9}{24} = \frac{288}{24} = 12
\][/tex]
Therefore, the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2 is 12 feet.
