Finding Altitude BD Length in Right Triangle

We are given a right-angled triangle ABC, where the right angle is at vertex B.

• The lengths of the legs are AB = 6 cm and BC = 8 cm.
• An altitude BD is drawn from vertex B to the hypotenuse AC.

Our goal is to find the length of the altitude BD.

Calculate Hypotenuse AC Length

First, we find the length of the hypotenuse AC using the Pythagoras theorem ($a^2 + b^2 = c^2$).

$AC^2 = AB^2 + BC^2$

Substitute the given values:

$AC^2 = 6^2 + 8^2$

$AC^2 = 36 + 64$

$AC^2 = 100$

$AC = \sqrt{100} = 10 \text{ cm}$

Calculate Altitude BD using Triangle Area

The area of a triangle can be calculated as $\frac{1}{2} \times \text{base} \times \text{height}$. We can calculate the area of triangle ABC in two ways:

1. Using the legs AB and BC as base and height:

   Area $= \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 6 \times 8 = 24 \text{ sq cm}$

2. Using the hypotenuse AC as the base and the altitude BD as the height:

   Area $= \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 10 \times BD$

Now, we equate the two expressions for the area:

$\frac{1}{2} \times 10 \times BD = 24$

$5 \times BD = 24$

Solve for BD:

$BD = \frac{24}{5}$

$BD = 4.8 \text{ cm}$

Final Answer

The length of the altitude BD is 4.8 cm.