Identifying the Longest Side in a Right Triangle

In mathematics, a right triangle or right-angled triangle is a triangle where one of the angles is exactly 90 degrees. Given a right triangle ABC with angle B 90 degrees, which side is the longest?

Understanding the Hypotenuse in Right Triangles

One fundamental property of right triangles is that the side opposite the 90-degree angle is always the longest. This side is known as the hypotenuse. In our example, since angle B is 90 degrees, the side opposite this angle, which is AC, is the hypotenuse and consequently, the longest side of the triangle.

The Pythagorean Theorem and Hypotenuse

The Pythagorean Theorem is a powerful tool for understanding the relationship between the sides of a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse (opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Formally, if a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse, then:

a2 b2 c2

Given the right triangle ABC, if we denote the sides as follows: AB as side a, BC as side b, and AC as side c, where side c is the hypotenuse, we can see that c is indeed the longest side due to the theorem.

Theorem 37 and the Longest Side

The statement, The side opposite the largest angle of a triangle is always the largest side, is a fundamental geometric principle. Applying this to our right triangle ABC, angle B being 90 degrees means it is the largest angle in the triangle. Therefore, the side opposite it, AC, must be the longest side, establishing it as the hypotenuse.

A related theorem, Theorem 37, asserts that in any triangle, the longest side is opposite the largest angle. Given that the sum of the angles in a triangle is 180 degrees, and angle B is 90 degrees, the other two angles, A and C, must be less than 90 degrees. This confirms that the side opposite the 90-degree angle, AC, is indeed the longest.

Conclusion

In summary, the longest side in a right triangle where angle B is 90 degrees is the hypotenuse AC. This is a direct consequence of the geometric properties of right triangles and the Pythagorean Theorem. Understanding these properties is essential for solving various mathematical problems and applications in fields such as engineering, architecture, and physics.

Frequently Asked Questions

Q: How do you find the hypotenuse of a right triangle?

A: The hypotenuse (the longest side) can be found using the Pythagorean Theorem, where c (hypotenuse) √(a2 b2).

Q: Can the longest side of a triangle be opposite the smallest angle?

A: No, the longest side of any triangle must always be opposite the largest angle. This is a fundamental geometric principle that applies to all types of triangles.

Q: What is the significance of the hypotenuse in right triangles?

A: The hypotenuse is significant because it is the longest side of a right triangle and its length can be used to calculate the lengths of the other two sides using the Pythagorean Theorem. It also plays a crucial role in many practical applications, such as in trigonometry and real-world problem-solving.