Questions: The tree in the illustration casts a shadow c=12 feet long while a person a=6 feet tall casts a shadow b=3 feet long. Find the height of the tree. 12 ft 1/24 ft 24 ft 6 ft 3/2 ft Through a point not on a given line, there is exactly one line parallel to the given line.

The tree in the illustration casts a shadow c=12 feet long while a person a=6 feet tall casts a shadow b=3 feet long. Find the height of the tree.
12 ft
1/24 ft
24 ft
6 ft
3/2 ft

Through a point not on a given line, there is exactly one line parallel to the given line.

Transcript text: The tree in the illustration casts a shadow $c=12$ feet long while a person $a=6$ feet tall casts a shadow $b=3$ feet long. Find the height of the tree. 12 ft $\frac{1}{24} \mathrm{ft}$ 24 ft 6 ft $\frac{3}{2} \mathrm{ft}$ Through a point not on a given line, there is exactly one line parallel to the given line.

Solution

Solution Steps

Step 1: Establish similar triangles

The man and his shadow form a right triangle. The tree and its shadow also form a right triangle. Because the sun creates the shadows at the same angle, these two triangles are similar. Therefore, the ratio of corresponding sides will be equal.

Step 2: Set up a proportion

We can set up a proportion relating the height of the man and his shadow to the height of the tree and its shadow:

\$\frac{\text{height of man}}{\text{man's shadow}} = \frac{\text{height of tree}}{\text{tree's shadow}}\$

Substituting the given values:

\$\frac{a}{b} = \frac{h}{c}\$

\$\frac{6}{3} = \frac{h}{12}\$

Step 3: Solve for the height of the tree

Cross-multiply to solve for _h_:

$3h = 6 \times 12$ $3h = 72$ $h = \frac{72}{3}$ $h = 24$