Question

For a project in his Geometry class, Chase uses a mirror on the ground to measure the height of his school’s football goalpost. He walks a distance of 7.25 meters from the goalpost, then places a mirror flat on the ground, marked with an X at the center. He then walks 6.55 more meters past the mirror, so that when he turns around and looks down at the mirror, 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 1.45 meters. How tall is the goalpost? Round your answer to the nearest hundredth of a meter.

Answer 1

We can solve the problem using similar triangles. Let's call the height of the goalpost "h".

When Chase looks down at the mirror, he sees the top of the goalpost and his own reflection. The line of sight from his eyes to the top of the goalpost makes an angle of θ with the horizontal, and the line of sight from his eyes to his reflection makes the same angle θ with the horizontal.

C

|\

| \

| \

| \

| \

| \

| \

| \

| \

|θ \

______|________\_________A

7.25m 6.55m

X

In the diagram, A is the top of the goalpost, C is Chase's eyes, X is the mirror on the ground, and θ is the angle of elevation. We want to find the height of the goalpost, h.

First, we need to find the length of CX. This can be found using the Pythagorean theorem:

CX² = AC² - AX²

CX² = (7.25m + 6.55m)² - (1.45m)²

CX² = 202.5m²

CX ≈ 14.23m

Next, we can use the fact that the triangles ACX and ABX are similar:

h / CX = 1.45m / (CX + 7.25m + 6.55m)

h / 14.23m = 1.45m / 20.03m

h ≈ 1.03m

Therefore, the height of the goalpost is approximately 1.03 meters, rounded to the nearest hundredth of a meter.