A side of an equilateral triangle is 20yd long. What is the area of the triangle? ( Picture Attached )

Question

anonymous · Answer

attachment

mathstudent55 · Answer

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

mathstudent55 · Answer

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.

mathstudent55 · Answer

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

mathstudent55 · Answer

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

mathstudent55 · Answer

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.

mathstudent55 · Answer

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.

mathstudent55 · Answer

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