Question

PLS SOMEONE HELP ME IM BEGGING

Answer 1

A = 16√3

Explanation:

This is a regular triangle, meaning each angle is 60° and all side lengths are equal. The apothem creates a 90° angle with a side.

The formula for area of a regular polygon given the apothem is:

A = (1/2)aP

where a is the apothem and P is the perimeter of the polygon.

Draw a line from the center point to an angle. Doing so creates a 30-60-90 triangle for which you have an angle and the apothem, which you can use to calculate the length of half a side.

tan(60) =
`(l/2)/((4√(3) )/(3) )`

So this says that the tangent if the 60° angle is equal to the half the length of one side, divided by the apothem.
Tangent → Opposite/Adjacent

So now we solve for l, the length of one side.

tan(60) =
`(l/2)/((4√(3) )/(3) )`

`√(3) = (l/2)/((4√(3) )/(3) ) \\ √(3) = (3(l/2))/(4√(3) )`

Cross-multiply.

`3(l/2) = 12\\l/2 = 4\\l = 8`

Based on the area formula, we need the perimeter, which is simply 3 * 8 because this is a regular triangle. The perimeter is 24. Now we have the necessary components to calculate the area of the triangle.

`A = (1/2)(24)((4√(3) )/(3) )`

A = 4 * 4
`√(3)`

A = 16√3