Question

find k.
write your answer as a whole number or a decimal. do not round.
k = meters

Explanation:

Step1: Determine similarity of triangles

Since \(\angle ABE=\angle ACD = 90^{\circ}\) and \(\angle A\) is common to both \(\triangle ABE\) and \(\triangle ACD\), by the AA (Angle - Angle) similarity criterion, \(\triangle ABE\sim\triangle ACD\).

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{AB}{AC}=\frac{BE}{CD}\).
We know \(AB = 8\) m, \(AC=24\) m, \(BE = k\), and \(CD = 45\) m.
Substituting the values into the proportion: \(\frac{8}{24}=\frac{k}{45}\).

Step3: Solve for \(k\)

Cross - multiply: \(24k=8\times45\).
First, calculate \(8\times45 = 360\). Then, \(k=\frac{360}{24}\).
Simplify \(\frac{360}{24}=15\).

Answer:

\(k = 15\) meters