Right Triangle Calculation: Finding Hypotenuse with Area 30 cm² and Base 5

To find the length of the hypotenuse of the triangle, let's proceed with the following steps:

• Step 1: Use the area formula to find the base.
• Step 2: Apply the Pythagorean Theorem to find the hypotenuse.

Step 1:
The formula for the area of a right-angled triangle is:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Given that the area is 30 cm², and one leg (the height) is 5 cm, we can find the base:

$\frac{1}{2} \times \text{base} \times 5 = 30$

Solve for $\text{base}$:

$\frac{1}{2} \times \text{base} \times 5 = 30$

Multiply both sides by 2:

$5 \times \text{base} = 60$

Divide by 5:

$\text{base} = 12 \, \text{cm}$

Step 2:
With base and height (legs of the triangle) known, apply the Pythagorean theorem to find the hypotenuse, $c$:

$c^2 = (\text{base})^2 + (\text{height})^2 = 12^2 + 5^2$

Calculate:

$c^2 = 144 + 25 = 169$

Take the square root to find $c$:

$c = \sqrt{169} = 13 \, \text{cm}$

Therefore, the length of the hypotenuse of the triangle is 13 cm.