Right Triangle

A right triangle is a triangle in which one of the angles is a right angle, i.e., 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can be written as:

c² = a² + b²

Properties of Right Triangle

• One angle is always 90 degrees.
• The sum of the other two angles is always 90 degrees.
• The sides are related by the Pythagorean theorem.

Special Right Triangles

There are two special types of right triangles: the 45-45-90 triangle and the 30-60-90 triangle. These triangles have specific ratios between their sides that make them useful for solving problems.

45-45-90 Triangle

In a 45-45-90 triangle, the two legs are congruent, and the length of the hypotenuse is equal to √2 times the length of a leg.

30-60-90 Triangle

In a 30-60-90 triangle, the length of the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.

Practice Problems

Solve the following problems:

1. In a right triangle, the lengths of the two legs are 3 and 4. Find the length of the hypotenuse.
2. Find the lengths of the sides of a 30-60-90 triangle if the shorter leg has a length of 6.

Conclusion

Understanding the properties of right triangles and the Pythagorean theorem is important for various mathematical and real-world applications. Practice problems and familiarize yourself with special right triangles to strengthen your understanding of the topic.