Question

John is trying to approximate the height of a nearby tree. He is 6 feet tall and measures his shadow to be 10.8 feet long. At the same time, the tree's shadow measures 63 feet.

(a) How tall is the tree? Draw a diagram.
(b) Your solution probably uses similar triangles. How do you know the triangles are similar?

Answer 1

Using the similar triangles method with the given measurements, John's height to shadow ratio is set equal to the tree's height to shadow ratio, resulting in the tree's estimated height of 35 feet. The similar triangles share an angle of elevation to the sun.

Step-by-step explanation:

To approximate the height of the tree using similar triangles, we start by setting up a proportion based on the idea that if two triangles are similar, their corresponding sides are proportional. John's height and shadow length form one triangle, and the tree's height and shadow form the larger, similar triangle.

Since John is 6 feet tall and has a shadow that is 10.8 feet long, and the tree's shadow is 63 feet long, we can set up the following proportion to solve for the tree's height (which we'll call ‘t’):

1. John's height / John's shadow length = Tree's height / Tree's shadow length
2. 6 feet / 10.8 feet = t / 63 feet
3. The proportion results in 6/10.8 = t/63.
4. To find t, we cross-multiply: 6 * 63 = 10.8 * t.
5. This results in 378 = 10.8t.
6. Dividing both sides by 10.8 gives us t = 35 feet.

Therefore, the tree is estimated to be 35 feet tall.

The triangles are known to be similar because they share the same angle of elevation to the sun, and their sides form the same ratios. This is known as the Angle-Angle (AA) similarity postulate, which states that two triangles are similar if they have two corresponding angles that are congruent.