## anonymous one year ago A side of an equilateral triangle is 20yd long. What is the area of the triangle? ( Picture Attached )

1. anonymous

2. mathstudent55

An equilateral triangle is also equiangular. All sides have length 20 yd, and all angles measure 60 deg. |dw:1437540579933:dw|

3. mathstudent55

To find the area, we can use the formula $$\Large A = \dfrac{bh}{2}$$ The base can be any side of the triangle. The height is the altitude.

4. mathstudent55

Let's use the bottom side as the base. Then we can draw the height as shown below. |dw:1437540791741:dw|

5. mathstudent55

Now we need the length of the height. |dw:1437540867820:dw|

6. mathstudent55

Each small triangle is a 30-60-90 triangle. We recal that a 30-60-90 triangle has sides in the ratio $$1 : \sqrt 3 : 2$$ The three sides of a 30-60-90 triangle are the two legs and the hypotenuse. The two legs form the right angle. The two legs are the short leg opposite the 30-deg angle, the long leg opposite the 60-deg angle. The hypotenuse is the longest side and is opposite the 90-deg angle. The ratio tells us that the hypotenuse is twice the length of the short leg. The long leg is $$\sqrt 3$$ times the length of the short leg.

7. mathstudent55

In the small triangle to the right, the short leg measures 10 yd. That means the long leg measures $$10 \sqrt 3~yd$$. The long leg of the small triangle to the right is also the height of the large triangle. Now we have the height we need to find the area.

8. mathstudent55

$$\large A = \dfrac{bh}{2} = \dfrac{20 ~yd \times 10\sqrt 3~yd}{2} = 100 \sqrt 3~yd^2$$