# Thread: Heights and distances

1. ## Heights and distances

A straight palm tree , 60 feet high , is broken by the wind but not completely separated , and its upper part meets the ground at an angle of 30 degrees . Find the distance of the point where the tree meets the ground , from the root , and also the height at which the tree is broken .

2. Ok, the first thing to note is that the ground and the two parts of the tree form a right triangle (see image).

Side a is the (still standing) lower part of the tree, side c is the (broken off) upper part of the tree, and side b is the ground.

Since we know that the total height of the tree was 60 ft, we can write:
$\displaystyle a+c = 60$ ft

We now have two unknown variables, but only equation. So we need one more equation. Luckily for a right triangle we have equations that relate angles and side lenghts. The sine of an angle in the triangle equals the opposing side divided by the hypotenuse side:
$\displaystyle \sin 30^{\circ} = \frac{a}{c}$
By combining the two equations you can now find a and c.

Afterwards you use Pythagoras' theorem to find the one remaining unknown side b:
$\displaystyle a^{2}+b^{2}=c^{2}$

3. it also helps to know that in a 30-60-90 triangle, c = 2a

4. Originally Posted by skeeter
it also helps to know that in a 30-60-90 triangle, c = 2a
Ah, true. That's simply what you get when you write out $\displaystyle \sin 30^{\circ}=\frac{a}{c}$.