Solving for the Altitude in a Right-Angled Triangle

This question asks us to find the length of the altitude drawn to the hypotenuse in a right-angled triangle. We are given the lengths of the two segments that the altitude divides the hypotenuse into.

We have a triangle ABC, which is right-angled at vertex A. An altitude AD is drawn from A, perpendicular to the hypotenuse BC. The point D lies on BC. We are given the lengths of the segments BD and DC.

• BD = 8 cm
• DC = 12.5 cm

We need to find the length of AD.

Applying the Geometric Mean Theorem for Altitude

In a right-angled triangle, when an altitude is drawn from the right angle to the hypotenuse, a special relationship exists between the altitude and the segments of the hypotenuse. This relationship is described by the Geometric Mean Theorem (also known as the Altitude Theorem).

The theorem states that the square of the altitude drawn to the hypotenuse is equal to the product of the lengths of the two segments into which the hypotenuse is divided by the altitude.

Mathematically, for $\triangle ABC$ right-angled at A, with AD $\perp$ BC:

$\text{AD}^2 = \text{BD} \times \text{DC}$

Calculating the Length of AD

Now, we can substitute the given values of BD and DC into the formula:

$\text{AD}^2 = 8 \text{ cm} \times 12.5 \text{ cm}$

Let's perform the multiplication:

$8 \times 12.5 = 8 \times \frac{125}{10} = 8 \times \frac{25}{2} = 4 \times 25 = 100$

So, we have:

$\text{AD}^2 = 100 \text{ cm}^2$

To find AD, we need to take the square root of 100:

$\text{AD} = \sqrt{100 \text{ cm}^2}$

$\text{AD} = 10 \text{ cm}$

Thus, the length of AD is 10 cm.

Summary of Calculation

The length of AD is 10 cm.

Revision Table: Key Concepts

Additional Information: Properties of Right Triangles with Altitude

Drawing an altitude to the hypotenuse of a right-angled triangle creates three similar triangles:

1. The original triangle (ABC)
2. The triangle formed by the altitude and one segment of the hypotenuse ($\triangle DBA$)
3. The triangle formed by the altitude and the other segment of the hypotenuse ($\triangle DAC$)

These similarities lead to other useful relationships, such as:

• $\text{AB}^2 = \text{BD} \times \text{BC}$ (Leg Theorem)
• $\text{AC}^2 = \text{DC} \times \text{BC}$ (Leg Theorem)
• $\text{AB}^2 + \text{AC}^2 = \text{BC}^2$ (Pythagorean Theorem, since $\text{BC} = \text{BD} + \text{DC}$)

In our case, $\text{BC} = \text{BD} + \text{DC} = 8 + 12.5 = 20.5$ cm. We could also use the leg theorem to find AB or AC if needed, but the altitude theorem directly gives us AD.