Question

a 12 - foot long ladder leans against the side of a house. if the ladder makes an angle of 60° with the ground, how far up the side of the house does the ladder reach?

6√3
√3
6
12

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the ladder is the hypotenuse ($c = 12$ feet), the height up the house is the side opposite the given angle ($\theta=60^{\circ}$). Use the sine function: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
So, $\sin60^{\circ}=\frac{h}{12}$, where $h$ is the height up the house.

Step2: Solve for \(h\)

Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, we substitute into the equation:
$\frac{\sqrt{3}}{2}=\frac{h}{12}$.
Cross - multiply: $h = 12\times\frac{\sqrt{3}}{2}$.
Simplify: $h = 6\sqrt{3}$.

Answer:

A. \(6\sqrt{3}\)