(Solved): #2 1) Using the Pythagorean Theorem, derive the area of an equilateral triangle with sides of length ...

1) Using the Pythagorean Theorem, derive the area of an equilateral triangle with sides of lengths \( a \). \[ \begin{array}{l} \left(\frac{a}{2}\right)^{2}+h^{2}=a^{2} \\ \frac{1}{4}(a)^{2}+h^{2}=a^{2} \\ h^{2}=a^{2}-\frac{1}{4}(a)^{2} \\ h^{2}=\frac{3}{4} a^{2} \\ h=\sqrt{\frac{3}{4} a^{2}} \\ h=\frac{1}{2} \sqrt{3 a} \end{array} \] 2) Now, consider a unit circle. a. Find the area of an equilateral triangle that is circumscribed by the unit circle. Center the base of the triangle at the origin. Let the right corner of the base be the point \( \left(\frac{a}{2}, 0\right) \) Let \( H \) represent the height of the equilateral triangle formed by the radii of the circle and the base of the triangle. b. Find the area of a square that is circumscribed by the unit cirdie. c. Find the area of a regular hexagon that is circumscribed by the unit circle.