Question

A dead tree was struck by lightening causing it to fall over at a point 10 ft up from its base. If the fallen treetop forms a 40° angle with the ground, about how tall was the tree originally?

Answer 1

25.6 ft

Explanation:

Although the problem here is listed under "Pythagorean theorem" you can't solve it by the Pythagorean theorem simply because you need to know the length of two sides of the right triangle formed by the broken tree and the stump.

But you can use trigonometry.

The broken tee and trunk form a right triangle with the ground.

The stump can be represented by the height of the triangle (10 ft.) while the fallen treetop can be represented by the hyptenuse of the triangle with the ground forming the base of the triangle.

So, we have a right triangle whose height is 10 ft. having an angle opposite the height of 40 degrees.

You are asked to find the original height of the tree so you need to find the length of the fallen treetop (the "hypotenuse") and then you'll add this to the tree stump (10 ft.) to find the original height of the tree. To find the length of the "hypotenuse", you can use the sin funtion of trigonometry because in a right triangle: Sin(A) = Opposite/Hypotenuse where the angle A (40 degrees)is the angle opposite the height (10 ft).

Sin%2840%29+=+10%2Fh where h is the hypotenuse. Solving for h, we get:

h+=+10%2FSin%2840%29

h+=+10%2F0.643

h+=+15.6ft.

Now add this to the 10-ft stump:

10+15.6 = 25.6 ft.

The tree was 25.6 ft originally.