Finding the Area of a Right-Angled Triangle

The question asks us to find the area of a right-angled triangle given its base and hypotenuse. Remember that the area of a triangle is calculated using the formula:

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

We are given:

• Base = 15 cm
• Hypotenuse = 17 cm

We have the base, but we need the height to calculate the area. We can find the height using the Pythagorean theorem, which applies to right-angled triangles. The theorem states:

$$a^2 + b^2 = c^2$$

Where 'a' and 'b' are the lengths of the two legs (base and height) and 'c' is the length of the hypotenuse.

Calculating the Triangle's Height

Let's apply the Pythagorean theorem:

• Let the base be $b = 15$ cm.
• Let the height be $h$.
• Let the hypotenuse be $c = 17$ cm.

Using the theorem:

$$b^2 + h^2 = c^2$$

Substitute the known values:

$$15^2 + h^2 = 17^2$$

Calculate the squares:

$$225 + h^2 = 289$$

Now, solve for $h^2$:

$$h^2 = 289 - 225$$

$$h^2 = 64$$

Find the height 'h' by taking the square root:

$$h = \sqrt{64}$$

$$h = 8 \text{ cm}$$

So, the height of the right-angled triangle is 8 cm.

Calculating the Area

Now that we have the base (15 cm) and the height (8 cm), we can calculate the area:

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

Area = $\frac{1}{2} \times 15 \text{ cm} \times 8 \text{ cm}$

Area = $\frac{1}{2} \times 120 \text{ cm}^2$

Area = $60 \text{ cm}^2$

Therefore, the area of the right-angled triangle is 60 cm².