Question

for a project in her geometry class, sofia uses a mirror on the ground to measure the height of her schools football goalpost. she walks a distance of 8.75 meters from the goalpost, then places a mirror flat on the ground, marked with an x at the center. she then walks 5.1 more meters past the mirror, so that when she turns around and looks down at the mirror, she can see the top of the goalpost clearly marked in the x. her partner measures the distance from her eyes to the ground to be 1.65 meters. how tall is the goalpost? round your answer to the nearest hundredth of a meter.

Explanation:

Step1: Set up proportion

Since the triangles are similar (by the law of reflection, the angles of incidence and reflection are equal, so the two right - angled triangles formed are similar). Let \(h\) be the height of the goalpost. The ratio of the height to the base of one triangle is equal to the ratio of the height to the base of the other triangle.
We have the proportion \(\frac{h}{8.75}=\frac{1.65}{5.1}\) (similar triangles: \(\text{height}_1/\text{base}_1=\text{height}_2/\text{base}_2\)).

Step2: Solve for \(h\)

Cross - multiply: \(h=\frac{1.65\times8.75}{5.1}\).
First, calculate \(1.65\times8.75 = 14.4375\).
Then, \(h=\frac{14.4375}{5.1}\approx2.83\).

Answer:

The height of the goalpost is approximately \(2.83\) meters.