Annie walks 15 feet away from her house and places a mirror on the ground. She backs 4 feet away from the mirror so that she can see the tip of the roof. Annie's eyes are 5 feet above the ground. Annie and the house are both perpendicular to the ground. The angles between the top of the house, the mirror, and the ground and between Annie's eyes, the mirror, and the ground are congruent as shown in the image below:
What is the height of the house? Show your work and explain your reasoning in complete sentences.
+248 Because the triangles are similar, 4:15 is in the same ratio as 5: height of the house.
To get from 4 to 5 you increast by 25%, increasing 15 by 25% means the height of the house is 18.75 feet tall.
-Annie's eyes are 5 feet above the ground, thus (Annie's is 5 ft) (To the left of the picture, opposite to the angle from mirror facing Annie)
and when she moved 15 feet away from her house, she backs 4 feet away from mirror Thus the distance between the angle facing Annie (let it be x)
and Annie is 4 feet
Using trigonometry,
\(tan(x)=\frac{5}{4}\) , It follows, \(x=tan^{-1}(\frac{5}{4})=51.34 \) degrees.
Since the 2 angles are congruent,
and the distance from the mirror and house 15 feet
again, using trigonometry to calculate the height of the house (Opposite side to the angle)
\(tan(51.34)=\frac{h}{15}\), where h is the height of the house
It follows, \(h=15tan(51.34)\)=18.75 correct to 2 decimal places feet.
