Question

Here is an equilateral triangle. The length of each side is

2 units. A height is drawn. In an equilateral triangle, a
line drawn from one corner to the center of the opposite
side represents the height.
1. Find the exact height.

2. Find the area of the equilateral triangle.

3. (Challenge) Using x for the length of each side in an
equilateral triangle, express its area in terms of x

Answer 1

An equilateral triangle is also an isosceles triangle.

In an isosceles triangle, the height is also the median line, which divide the line segment into 2 equal parts.

Therefore, 2 right-angled triangles are formed with the hypotenuse's length being 2 units and the shorter side being 1 unit.

We can calculate the height using Pythagorean's theorem :
`\sqrt{2^(2)-1^(2) } = √(3)`

2. The area of the equilateral triangle =
`(2√(3) )/(2) = √(3)` (unit square)

3. Again, we calculate the height if each side is x using the method above :
`\sqrt{x^(2)- (x^(2) )/(2^(2) ) } = \sqrt{(3x^(2) )/(4) } = √(3) (x)/(2)`

-> Area =
`√(3)(x)/(2) x:2` =
`√(3) (x^(2) )/(4)` (not inside root)

Answer 2

The area of the equilateral triangle is sqrt(3) square units.

Explanation:

• In an equilateral triangle, the height is also the altitude, and it bisects the opposite side into two equal parts. Therefore, the height of an equilateral triangle with side length 2 units can be found using the Pythagorean Theorem as follows:

h^2 = 2^2 - (2/2)^2

h^2 = 4 - 1

h^2 = 3

h = √3 units (exact height)

• The area of an equilateral triangle can be found using the formula:

Area = (sqrt(3)/4) x (side length)^2

Plugging in the values, we get:

Area = (sqrt(3)/4) x 2^2

Area = (sqrt(3)/4) x 4

Area = sqrt(3) square units

Therefore, the area of the equilateral triangle is sqrt(3) square units.

• The area of an equilateral triangle with side length x can be expressed as:

Area = (sqrt(3)/4) x (x)^2

Area = (x^2 * sqrt(3))/4 square units.