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Triangle 1 and triangle 2 are similar right triangles formed from a ladder leaning against a building.

[tex]\[
\begin{tabular}{|c|c|}
\hline
\textbf{Triangle 1} & \textbf{Triangle 2} \\
\hline
\begin{tabular}{l}
The distance, along the ground, from the bottom of the ladder to the building is 24 feet. \\
The distance from the bottom of the building to the point where the ladder is touching the building is 32 feet.
\end{tabular}
&
\begin{tabular}{l}
The distance, along the ground, from the bottom of the ladder to the building is 9 feet. \\
The distance from the bottom of the building to the point where the ladder is touching the building is unknown.
\end{tabular}
\\
\hline
\end{tabular}
\][/tex]

Determine the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2:

A. 8 feet
B. 12 feet
C. 18 feet
D. 27 feet



Answer :

GinnyAnswer
Alright, let's find the required height for triangle 2 given this problem involving similar triangles formed by a ladder leaning against a building.

Since the triangles are similar, the ratio of the corresponding sides will be equal. In triangle 1:
- The distance along the ground from the base of the ladder to the building is 24 feet.
- The distance from the base of the building to the point where the ladder touches the building (height) is 32 feet.

For triangle 2:
- The distance along the ground from the base of the ladder to the building is 9 feet.
- We need to determine the height from the base of the building to the point where the ladder touches the building.

We know the ratio for triangle 1 is:
[tex]\[ \text{Ratio} = \frac{\text{Height in triangle 1}}{\text{Distance along the ground in triangle 1}} = \frac{32}{24} \][/tex]

Using this ratio for triangle 2:
[tex]\[ \text{Height in triangle 2} = \text{Ratio} \times \text{Distance along the ground in triangle 2} = \frac{32}{24} \times 9 \][/tex]

Simplify the ratio:
[tex]\[ \frac{32}{24} = \frac{4}{3} \][/tex]

Now, multiply the simplified ratio by the distance along the ground in triangle 2:
[tex]\[ \text{Height in triangle 2} = \frac{4}{3} \times 9 = 12 \text{ feet} \][/tex]

Therefore, the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2 is:
[tex]\[ \boxed{12 \text{ feet}} \][/tex]

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