Area Triangle Bec 0F3679

1. **Problem Statement:** We are given a rhombus $ABCD$ with side length $15$ cm and a segment $EC = 12$ cm. We need to find the area of the right triangle $BEC$. 2. **Understanding the figure:** Since $ABCD$ is a rhombus, all sides are equal: $AB = BC = CD = DA = 15$ cm. $\triangle BEC$ is right angled at $E$, with $EC = 12$ cm vertical and $BE$ horizontal. 3. **Find length $BE$:** Since $BE$ is horizontal and $EC$ is vertical, $\triangle BEC$ is a right triangle with legs $BE$ and $EC$ and hypotenuse $BC$. Using the Pythagorean theorem: $$BC^2 = BE^2 + EC^2$$ $$15^2 = BE^2 + 12^2$$ $$225 = BE^2 + 144$$ $$BE^2 = 225 - 144 = 81$$ $$BE = \sqrt{81} = 9 \text{ cm}$$ 4. **Calculate the area of $\triangle BEC$:** Area of a right triangle is: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ Here, base $= BE = 9$ cm and height $= EC = 12$ cm. $$\text{Area} = \frac{1}{2} \times 9 \times 12 = \frac{1}{2} \times 108 = 54 \text{ cm}^2$$ **Final answer:** $$\boxed{54 \text{ cm}^2}$$