Right Triangle Hypotenuse 2Ea1C8

1. **State the problem:** We have a right-angled triangle with base $21cm$ and area $147cm^2$. We need to find the length of the longest side (the hypotenuse). 2. **Recall the formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Use the given values to find the height:** $$147 = \frac{1}{2} \times 21 \times h$$ 4. **Solve for $h$:** $$147 = \frac{21}{2} \times h$$ Multiply both sides by $2$: $$2 \times 147 = 21 \times h$$ $$294 = 21 \times h$$ Divide both sides by $21$: $$h = \frac{294}{21}$$ Show cancellation: $$h = \frac{\cancel{294}}{\cancel{21}} = 14$$ So, the height $h = 14cm$. 5. **Identify the sides of the right triangle:** - Base $b = 21cm$ - Height $h = 14cm$ - Hypotenuse $c = ?$ 6. **Use the Pythagorean theorem to find the hypotenuse:** $$c^2 = b^2 + h^2$$ $$c^2 = 21^2 + 14^2$$ $$c^2 = 441 + 196$$ $$c^2 = 637$$ 7. **Calculate $c$:** $$c = \sqrt{637}$$ Approximate: $$c \approx 25.24cm$$ **Final answer:** The length of the longest side (hypotenuse) is approximately $25.24cm$.