Calculating the Altitude on the Hypotenuse of a Right-Angled Triangle

In geometry, understanding the dimensions and properties of shapes is crucial. One specific scenario involves right-angled triangles, where finding the altitude on the hypotenuse can be essential for various applications, from engineering to everyday problem-solving. This article explores the methodology and provides examples to help you calculate the altitude on the hypotenuse of a right-angled triangle.

Understanding the Basics

A right-angled triangle is a triangle with one of its angles being exactly 90 degrees. The side opposite the right angle is called the hypotenuse, which is the longest side of the triangle. The two other sides are called the legs or the base and the altitude.

Problems and Solutions

Let's consider a problem where we need to find the altitude on the hypotenuse of a right-angled triangle given certain dimensions. This involves using the formulas and principles of geometry, particularly the area of the triangle and the application of Pythagoras' theorem.

First Example

The area of a right-angled triangle is 150 cm2 and one of the sides containing the right angle is 4 cm. We need to find the approximate length, in cm, of the altitude on the hypotenuse.

Given:

Area, A 150 cm2 One leg, AB 4 cm

From the area formula for a right-angled triangle, we have:

1/2 * AB * BC 150

Solving for BC:

1/2 * 4 * BC 150

BC 75 cm

We then find the hypotenuse AC by using the Pythagorean theorem:

AC √(AB2 BC2) √(42 752) √(16 5625) √5641

Now, using the area formula again for the altitude on the hypotenuse, L:

1/2 * L * AC 150

L 300 / √5641 ≈ 3.994 ≈ 4 cm

Second Example

The area of a right-angled triangle is 24 cm2 and one of the sides containing the right angle is 6 cm. We need to find the length of the altitude on the hypotenuse.

Given:

Area, A 24 cm2 One leg, AB 6 cm

From the area formula for a right-angled triangle, we have:

1/2 * AB * BC 24

Solving for BC:

1/2 * 6 * BC 24

BC 8 cm

Now, we find the hypotenuse AC using the Pythagorean theorem:

AC √(AB2 BC2) √(62 82) √(36 64) √100 10 cm

Using the area formula again for the altitude on the hypotenuse, L:

1/2 * L * AC 24

1/2 * L * 10 24

L 24 * 2 / 10 4.8 cm

Third Example

Another problem involves finding the altitude on the hypotenuse of a right-angled triangle with given areas and side lengths.

Given:

Area, A 20 cm2 One leg, AB 4 cm Let BC be the other leg. Using the area formula for a right-angled triangle:

1/2 * 4 * BC 20

Solving for BC:

BC 10 cm

Now, finding the hypotenuse AC:

AC √(AB2 BC2) √(42 102) √(16 100) √116 2√29 cm

Using the area formula again for the altitude on the hypotenuse, L:

1/2 * 2√29 * L 20

L 20 * 2 / 2√29 20√29 / 29 ≈ 3.71 cm

Conclusion

Understanding how to calculate the altitude on the hypotenuse of a right-angled triangle is crucial for many practical applications, including architectural designs, land surveying, and building construction. By using the formulas derived from the area of the triangle and the Pythagorean theorem, you can effectively solve such problems.