Answer :
To solve this problem, we can use similar triangles. Similar triangles have the same shape but different sizes, and corresponding angles are equal with the sides in the same ratio.
Let's consider the situation described where there are two similar triangles:
1. The small triangle that is formed by Pete, the point on the ground where he is looking into the mirror, and the point at the top of his head.
2. The large triangle that is formed by the point on the ground where the mirror is located, the base of the tree, and the top of the tree.
Since Pete can just see the top of the tree in the mirror, the angles at which Pete's eyes are looking are the same as the angles formed by the imaginary line between the mirror and the top of the tree. This creates two similar right-angled triangles.
Let's denote the height of the tree as \( T \). In the small triangle, the height (opposite side) is 4 feet (Pete's height) and the base (adjacent side to the angle of interest) is 4 feet (the distance Pete stands from the mirror). In the large triangle, the height (opposite side) is \( T \) and the base (adjacent side) is the distance from the mirror to the base of the tree, which is 16 feet.
The ratios of the sides of similar triangles are equal. So, we can set up the equation:
\[ \frac{\text{Height of Pete}}{\text{Distance of Pete from the mirror}} = \frac{\text{Height of the tree}}{\text{Distance of the mirror from the tree}} \]
Substituting the given numbers, we have:
\[ \frac{4}{4} = \frac{T}{16} \]
Solving for \( T \) by cross-multiplying:
\[ 4 \times 16 = T \times 4 \]
\[ 64 = 4T \]
Divide both sides by 4:
\[ \frac{64}{4} = T \]
\[ 16 = T \]
So, the height of the tree is 16 feet.
