Once drawing the pentagon with a radius of 2, how would you figure out the side length when only knowing that the radius is 2 and when the apothem is placed it splits into a 36-60-90 triangle?
To find the side length of a pentagon with a given radius of 2, and when the apothem splits into a 36-60-90 triangle, you can apply some geometry and trigonometry principles.
Let's start by understanding the properties of a regular pentagon:
1. A regular pentagon has five equal sides (all sides have the same length) and five equal angles (each angle measures 108 degrees).
2. The apothem is a line segment drawn from the center of the pentagon to the midpoint of any side, forming right angles with that side.
In your case, when the apothem is drawn, it splits the pentagon into two congruent triangles, one of which is a 36-60-90 triangle. In a 36-60-90 triangle, the side lengths ratios are:
- The side opposite the 36-degree angle is x.
- The side opposite the 60-degree angle is x√3.
- The side opposite the 90-degree angle is 2x.
We know that the apothem splits the pentagon into two congruent triangles, and since we have a 36-60-90 triangle, we can equate the apothem to x (the side opposite the 36-degree angle).
Given that the radius of the pentagon is 2, it means the apothem (x) is also 2 units.
Now, since the side opposite the 90-degree angle in the 36-60-90 triangle is 2x, we can multiply the apothem (x = 2) by 2:
2 * 2 = 4.
Therefore, the side length of the pentagon is 4 units.