Question

I will award 50 points and brainliest to the best proof with explanation :D

Answer 1

Step-by-step explanation:

(You might need to draw the triangle on a piece of paper to follow)

First I tried to find the height in terms of a.  

If you draw the height in, the triangle is cut in half. That half triangle is a right triangle.  

Using the Pythagoras theorem you can find out the height. Since you know that the hypotenuse is also a (because it's an equilateral triangle) the equation would be:

(a/2)² + h² = a²

Assuming that h = height

(It's a/2 because when you cut the triangle in half, the base of the right triangle became a/2)

This equation is simplified:

a²/4 + h² = a²

a² + 4h = 4a²

4h = 3a²

h² = 3a²/4

Now, remeber that everything is squared so it should be

h = a√3/2

Now we have the height. To get the area we need to multiply it by the base, which is a, and then divide it by 2.

a√3/2 × a ÷ 2

a²√3/2 ÷ 2

a²√3/4

And there you go. Thats the area.

Answer 2

Step-by-step explanation:

(You might need to draw the triangle on a piece of paper to follow)

First I tried to find the height in terms of a.  

If you draw the height in, the triangle is cut in half. That half triangle is a right triangle.  

Using the Pythagoras theorem you can find out the height. Since you know that the hypotenuse is also a (because it's an equilateral triangle) the equation would be:

(a/2)² + h² = a²

Assuming that h = height

(It's a/2 because when you cut the triangle in half, the base of the right triangle became a/2)

This equation is simplified:

a²/4 + h² = a²

a² + 4h = 4a²

4h = 3a²

h² = 3a²/4

Now, remeber that everything is squared so it should be

h = a√3/2

Now we have the height. To get the area we need to multiply it by the base, which is a, and then divide it by 2.

a√3/2 × a ÷ 2

a²√3/2 ÷ 2

a²√3/4

And there you go. Thats the area.