Question

a painter needs to measure the height from the ground to the base of a 2nd story window. he props his ladder against the base of the window. he finds the height from one rung of the ladder to the point on the ground below it, and the distance from that point to the base of the ladder. this is shown in the figure. note: figure is not drawn to scale. if xy = 8 inches, wy = 24 inches, and xz = 48 inches, the height from the ground to the base of the window is inches.

Explanation:

Step1: Identify Similar Triangles

The triangles \( \triangle WXY \) and \( \triangle VXZ \) are similar (right triangles with common angle at \( X \), so by AA similarity).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So \( \frac{WY}{VZ} = \frac{XY}{XZ} \).
We know \( WY = 24 \), \( XY = 8 \), \( XZ = 48 \). Let \( VZ = h \) (height we need to find).
Substitute values: \( \frac{24}{h} = \frac{8}{48} \).

Step3: Solve for \( h \)

Cross - multiply: \( 8h = 24\times48 \).
First, calculate \( 24\times48 = 1152 \).
Then, \( h=\frac{1152}{8}=144 \).

Answer:

144