For a project in her Geometry class, Deepa uses a mirror on the ground to measure the height of her school’s football goalpost. She walks a distance of 9.35 meters from the goalpost, then places a mirror on flat on the ground, marked with an X at the center. She then steps 6.95 meters to the other side of the mirror, until 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 1.35 meters. How tall is the goalpost? Round your answer to the nearest hundredth of a meter.

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oeerivona oeerivona · Answer 1 · 2021-12-04T09:09:50+01:00

Deepa makes use of the geometric property of similar triangles formed by

incident and reflected rays to determine the height of the goalpost.

Reasons:

When an object is reflected on a mirror, the angle of incidence, θ₁ is equal

to the angle of reflection, θ₂.

θ₁ = θ₂

Given that the angle, ∅₁, the incident ray of light and the angle, ∅₂, the

reflected light make with horizontal are both complementary angles, to the

angle of incident and reflection, respectively, we have;

θ₁ + ∅₁ = θ₂ + ∅₂

θ₁ = θ₂

Therefore, by subtraction property of equality, we have;

∅₁ = ∅₂

The vertical line from the top of the goalpost to the base of the goalpost

and the the vertical line from Deepa's eyes to the ground on which her feet

is standing are both perpendicular to the ground, therefore, the light from

the top of the goalpost to the mirror and to her eyes form similar triangles

by Angle Angle similarity postulate, which gives;

[tex]\displaystyle \frac{6.95}{9.35} = \frac{1.35}{The \ height \ of \ the \ goalpost}[/tex]

6.95 × Height of the goalpost = 1.35 × 9.35

[tex]\displaystyle Height \ of \ the \ goalpost = \frac{1.35 \ m \times 9.35 \ m}{6.95 \ m} \approx 1.82 \ m[/tex]

The height of the goalpost is approximately 1.82 meters.

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