Find the area of an equilateral triangle as a function of x, where 2x is equal to the side length of triangle?

Answer 1

Drop a line straight down from the top of the triangle to the base. That's the height of the triangle.

You have a right triangle now, with the hypotenuse = 2x (one side) and the shorter side = x, and the angle between is 60 degrees.

Therefore the height of the triangle = sqrt(4x^2 - x^2) = xsqrt(3).

The area of the triangle is 1/2 the base (2x) times the height.

A(x) = 1/2(2x)(xsqrt(3)) = x^2(sqrt(3))

Answer 2

The area of a triangle is equal to 1/2 times its height times its base. The base is obviously going to be 2x. If you split up the equalateral triangle into two pieces going down the middle you get two right triangles sharing the height as their common side. Each right triangle would have x as their short side and 2x as their hypotenuse. You can find the length of the third side (the height) using the Pythagorean theorem...

x^2 + h^2 = (2x)^2 = 4*x^2
h^2 = 3*x^2
h = sqrt(3)*x

So the height is square root of 3 times x.

The area would then be...

A = 1/2bh = 1/2(2x)[sqrt(3)*x] = sqrt(3)*x^2

Answer 3

danny has given you a simple and correct answer.

Here's an explanation of where it came from:
If you're studying geometry, you should memorize the following formula for the area of an equilateral triangle:

A = (s^2) (sqrt(3)) / 4
where s is the length of a side of the triangle

In your example, s = 2x. So danny has simply substituted 2x for s in the above equation. Once you simplify it, you should find that the answer is "x square times the square root of 3."

Answer 4

A= (1/2*x)*(sqrt(3)*x)

start by drawing an equalateral triangle and label each side 2x

we know that the area of a triangle is 1/2*base*height

finding the height is the tricky part, use the pythagorean theorem a^2+b^2=c^2 and fill in the information

the hypotenuse is c, which in this case is 2x, the length of the base is 1/2 one of the sides which is x, so plug it in to get x^2+height^2 =(2x)^2

when you solve for height you get height^2=4x^2-x^2
height =sqrt(3)*x

plug this in again

area = 1/2*base*height

area =1/2*(2x)* (sqrt(3)*x)

Answer 5

The area can be calculated by : bh/2

We know that b = 2x

In order to find h we divide de triangle in two rectangle traingle where the catets are h and x and the hypot^2= 2x (According to Pitagoras)
4x^2 = x^2 + h^2
h= sqr(3) x.... sqr: square root

Now the Area = 2x . sqr(3) x/2 = x^2 sqr(3) ... this the formula you are looking for

Answer 6

2x is equal to the opposite site of this triangle since the opposite site of a equilateral triangle are equal.

Answer 7

2x = length of side

area = (2x)^2 (squareroot of 3) ALL DIVIDED BY 4
Then you simplify it