Question

for a project in his geometry class, david uses a mirror on the ground to measure the height of his schools football goalpost. he walks a distance of 14.75 meters from the goalpost, then places a mirror flat on the ground, marked with an x at the center. he then steps 1.45 meters to the other side of the mirror, until he can see the top of the goalpost clearly marked in the x. his partner measures the distance from his eyes to the ground to be 1.55 meters. how tall is the goalpost? round your answer to the nearest hundredth of a meter.

Explanation:

Step1: Set up the proportion

Since the triangles are similar (by the AA similarity criterion, as the angles of incidence and reflection are equal and the right angles are equal), we can set up the proportion \(\frac{\text{height of person}}{\text{distance from person to mirror}}=\frac{\text{height of goalpost}}{\text{distance from goalpost to mirror}}\). Let \(h\) be the height of the goalpost. So, \(\frac{1.55}{1.45}=\frac{h}{14.75}\).

Step2: Solve for \(h\)

Cross - multiply: \(1.45h = 1.55\times14.75\). First, calculate \(1.55\times14.75=1.55\times(14 + 0.75)=1.55\times14+1.55\times0.75 = 21.7+1.1625=22.8625\). Then, \(h=\frac{22.8625}{1.45}\).
\(h=\frac{22.8625}{1.45}=15.767241\cdots\)

Answer:

\(15.77\) meters