The base and altitude of at right angled triangle are 12 cm and 5 cm respectively. The perpendicular distance of its hypotenuse from the opposite vertex is

Options

A. `4(4)/(13)cm`
B. `4(8)/(13) cm`
C. 5 cm
D. 7 cm

Answer

To find the perpendicular distance of the hypotenuse from the opposite vertex in a right-angled triangle with a base of 12 cm and an altitude of 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle Dimensions**:
- Base (AB) = 12 cm
- Altitude (BC) = 5 cm

2. **Calculate the Hypotenuse (AC)**:
- Using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
- Substitute the values:
\[ AC^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
- Therefore, the hypotenuse (AC) is:
\[ AC = \sqrt{169} = 13 \text{ cm} \]

3. **Calculate the Area of the Triangle**:
- The area (A) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Substitute the values:
\[ A = \frac{1}{2} \times 12 \times 5 = 30 \text{ cm}^2 \]

4. **Use the Area to Find the Perpendicular Distance from the Hypotenuse**:
- The area can also be expressed in terms of the hypotenuse and the perpendicular distance (h) from the opposite vertex (C) to the hypotenuse (AB):
\[ A = \frac{1}{2} \times \text{hypotenuse} \times \text{perpendicular distance} \]
- Rearranging gives:
\[ \text{perpendicular distance} = \frac{2A}{\text{hypotenuse}} \]
- Substitute the values:
\[ \text{perpendicular distance} = \frac{2 \times 30}{13} = \frac{60}{13} \approx 4.615 \text{ cm} \]

5. **Final Answer**:
- The perpendicular distance from the opposite vertex to the hypotenuse is approximately \( \frac{60}{13} \) cm or about 4.615 cm.