Area of a right-angled triangle is `30 cm^2`. If its smallest side is 5 cm, then its hypotenuse is

Options

A. 14cm
B. 13 cm
C. 12 cm
D. 11 cm

Answer

To find the hypotenuse of a right-angled triangle when the area and one side are given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the area of a right-angled triangle**:
The area \( A \) of a right-angled triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{perpendicular} \]
Here, we know the area is \( 30 \, \text{cm}^2 \).

2. **Assign the smallest side**:
Since the smallest side is given as \( 5 \, \text{cm} \), we can assume this is the base. Let's denote:
\[ \text{base} = 5 \, \text{cm} \]
Let the perpendicular side be \( P \).

3. **Set up the equation for the area**:
Plugging the values into the area formula:
\[ 30 = \frac{1}{2} \times 5 \times P \]

4. **Solve for the perpendicular side \( P \)**:
Multiply both sides by \( 2 \):
\[ 60 = 5 \times P \]
Now, divide both sides by \( 5 \):
\[ P = \frac{60}{5} = 12 \, \text{cm} \]

5. **Use the Pythagorean theorem**:
In a right-angled triangle, the Pythagorean theorem states:
\[ \text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2 \]
Substituting the known values:
\[ h^2 = 5^2 + 12^2 \]

6. **Calculate \( 5^2 \) and \( 12^2 \)**:
\[ 5^2 = 25 \quad \text{and} \quad 12^2 = 144 \]
Therefore:
\[ h^2 = 25 + 144 = 169 \]

7. **Find the hypotenuse \( h \)**:
Taking the square root of both sides:
\[ h = \sqrt{169} = 13 \, \text{cm} \]

### Final Answer:
The hypotenuse of the triangle is \( 13 \, \text{cm} \).