12. Find the side length and area of an equilateral triangle with apothem 3 cm .

1. The apothem \( r \) of an equilateral triangle with side length \( a \) is given by \[ r = \frac{\sqrt{3}}{6}a. \] Since \( r = 3 \) cm, we have \[ 3 = \frac{\sqrt{3}}{6}a. \] 2. Multiply both sides by 6 to isolate \( a \): \[ 18 = a\sqrt{3}. \] 3. Solve for \( a \) by dividing both sides by \( \sqrt{3} \): \[ a = \frac{18}{\sqrt{3}}. \] Rationalize the denominator: \[ a = \frac{18\sqrt{3}}{3} = 6\sqrt{3} \text{ cm}. \] 4. The area \( A \) of an equilateral triangle is given by \[ A = \frac{\sqrt{3}}{4}a^2. \] Substitute \( a = 6\sqrt{3} \): \[ A = \frac{\sqrt{3}}{4}\left(6\sqrt{3}\right)^2. \] 5. Compute \(\left(6\sqrt{3}\right)^2\): \[ \left(6\sqrt{3}\right)^2 = 36 \times 3 = 108. \] Thus, \[ A = \frac{\sqrt{3}}{4} \times 108. \] 6. Simplify the area: \[ A = \frac{108\sqrt{3}}{4} = 27\sqrt{3} \text{ cm}^2. \] The side length is \(6\sqrt{3}\) cm and the area is \(27\sqrt{3}\) cm\(^2\).