Find the area of the right-angled triangle with hypotenuse 40 cm and one of the other two sides 24 cm.

Answer

To find the area of the right-angled triangle with a hypotenuse of 40 cm and one side of 24 cm, we can follow these steps:

### Step 1: Identify the triangle and given values
Let’s label the triangle as ABC, where:
- AC is the hypotenuse = 40 cm
- AB is one of the sides = 24 cm
- BC is the other side, which we need to find.

### Step 2: Use the Pythagorean theorem
According to the Pythagorean theorem, in a right-angled triangle:
\[ AC^2 = AB^2 + BC^2 \]

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ 40^2 = 24^2 + BC^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 1600 = 576 + BC^2 \]

### Step 5: Solve for BC^2
Now, isolate \( BC^2 \):
\[ BC^2 = 1600 - 576 \]
\[ BC^2 = 1024 \]

### Step 6: Find the length of BC
Taking the square root of both sides to find BC:
\[ BC = \sqrt{1024} \]
\[ BC = 32 \text{ cm} \]

### Step 7: Calculate the area of the triangle
The area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take BC as the base and AB as the height:
\[ A = \frac{1}{2} \times BC \times AB \]
\[ A = \frac{1}{2} \times 32 \times 24 \]

### Step 8: Perform the multiplication and division
Calculating the area:
\[ A = \frac{1}{2} \times 32 \times 24 = 16 \times 24 = 384 \text{ cm}^2 \]

### Final Answer
The area of the right-angled triangle is \( 384 \text{ cm}^2 \).

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