Find the area of a right angled triangle whose height is 3 cm and hypotenuse 5 cm.

Options

A. `5 cm^(2)`
B. `6 cm^(2)`
C. `8 cm^(2)`
D. `10 cm^(2)`

Answer

To find the area of a right-angled triangle with a height of 3 cm and a hypotenuse of 5 cm, we can follow these steps:

### Step 1: Identify the given values
- Height (perpendicular) = 3 cm
- Hypotenuse = 5 cm

### Step 2: Use the Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]

Substituting the known values:
\[ 5^2 = 3^2 + \text{Base}^2 \]

### Step 3: Calculate the squares
Calculating the squares:
- \( 5^2 = 25 \)
- \( 3^2 = 9 \)

So we have:
\[ 25 = 9 + \text{Base}^2 \]

### Step 4: Rearrange the equation to find the base
Rearranging the equation gives:
\[ \text{Base}^2 = 25 - 9 \]
\[ \text{Base}^2 = 16 \]

### Step 5: Calculate the base
Taking the square root of both sides:
\[ \text{Base} = \sqrt{16} = 4 \text{ cm} \]

### Step 6: Calculate the area of the triangle
The area \( A \) of a right-angled triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]

Substituting the values we found:
\[ A = \frac{1}{2} \times 4 \times 3 \]

### Step 7: Perform the multiplication
Calculating the area:
\[ A = \frac{1}{2} \times 12 = 6 \text{ cm}^2 \]

### Final Answer
The area of the right-angled triangle is **6 cm²**.

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