Question

Tanya wants to find the height of the tree. She walks away from the base of the tree so the tip of her shadow coincides with the tip of the tree's shadow at point C. BE is two and a half times EC. Tanya is 5 feet, 3 inches tall. How tall is the tree in feet?

Answer 1

Refer to the diagram attached.

TRIANGLE ABC
The height of the tree is the segment AB. The shadow of the tree is BC.

TRIANGLE DEC
Tanya's height is 5 feet 3 inches (since there are 12 inches in one foot that is (5)(12)+3 = 63 inches).
Tanya's shadow is EC

Since both triangles are right triangles (the both have a right angle) and they share the angle at C, they are similar triangles. That means that their corresponding angles are congruent and their corresponding sides are in proportion.

We can set up the following proportion:
(tree height / tree shadow) = (Tanya's height / Tanya's shadow)

BE is 2.5 times EC. This means that BE is (2.5)(EC). It also means that BC = 2.5EC + EC = 3.5EC

The shadow of the tree is 3.5 times Tanya's shadow. That means that the height of the tree must be 3.5 times Tanya's height.

The height of the tree is (3.5)(63) = 220.5 inches. If we divide this by 12 we get 18.375 which means 18 feet and some inches.  (18)(12)=216 and the tree is 220.5. That means the tree is 18 feet and 4.5 inches.