Question

Standing 4 feet from a mirror resting on Rat ground, Joseph, whose eye height is 5 feet, a inches can see the reflection of the top of a tree. He measures the mirror to be 24 feet from the tree. How tall is the tree?

Answer 1

Based on the angle, angle (AA) similarity theorem, the tree is 34.5 feet tall.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Furthermore, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the angle, angle (AA) similarity theorem, we can logically deduce the following proportional sides:

x/5.75 = 24/4

x/5.75 = 6

x = 6 × 5.75

x = 34.5 feet.

Answer 2

Similar Triangles

Similar triangles can be identified because their corresponding side lengths are in the same proportion.

The image shows a situation where Joseph, the mirror, and the tree form two similar triangles, assuming the angle of elevation from the mirror is the same in both directions, as shown.

This means the proportions of the heights of the tree and Joseph have the same ratio as their distances to the mirror, that is:

`(x)/(5.75)=(24)/(4)`
`(x)/(5.75)=6`

Multiplying by 5.75:

x = 5.75*6 = 34.5

The tree is 34.5 feet tall