Understanding the Pythagorean Theorem

Oct 13, 2025 by miodeya.com 54 views

Which of these four sets of side lengths will form a right triangle?

A right triangle is a triangle in which one of the angles is a right angle, i.e., 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right triangle. The theorem states that 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), i.e., a^2 + b^2 = c^2.

Understanding the Pythagorean Theorem

The Pythagorean theorem is a powerful tool for determining whether a given set of side lengths can form a right triangle. To apply the theorem, we need to identify the lengths of the three sides of the triangle. Let's consider the four sets of side lengths given in the problem:

Set 1: 10 cm, 26 cm, 24 cm
Set 2: 4 in, 2 in, 6 in
Set 3: 14 cm, 48 cm, 50 cm
Set 4: 9 in, 12 in, 15 in

We will use the Pythagorean theorem to determine which of these sets of side lengths can form a right triangle.

Applying the Pythagorean Theorem

To apply the Pythagorean theorem, we need to calculate the sum of the squares of the lengths of the two shorter sides and compare it with the square of the length of the longest side. If the sum of the squares of the two shorter sides is equal to the square of the length of the longest side, then the triangle is a right triangle.

Let's apply the theorem to each of the four sets of side lengths:

Set 1: 10 cm, 26 cm, 24 cm

Calculate the sum of the squares of the two shorter sides: 10^2 + 26^2 = 100 + 676 = 776
Calculate the square of the length of the longest side: 24^2 = 576
Compare the two values: 776 ≠ 576

Since the sum of the squares of the two shorter sides (776) is not equal to the square of the length of the longest side (576), Set 1 cannot form a right triangle.

Set 2: 4 in, 2 in, 6 in

Calculate the sum of the squares of the two shorter sides: 4^2 + 2^2 = 16 + 4 = 20
Calculate the square of the length of the longest side: 6^2 = 36
Compare the two values: 20 ≠ 36

Since the sum of the squares of the two shorter sides (20) is not equal to the square of the length of the longest side (36), Set 2 cannot form a right triangle.

Set 3: 14 cm, 48 cm, 50 cm

Calculate the sum of the squares of the two shorter sides: 14^2 + 48^2 = 196 + 2304 = 2500
Calculate the square of the length of the longest side: 50^2 = 2500
Compare the two values: 2500 = 2500

Since the sum of the squares of the two shorter sides (2500) is equal to the square of the length of the longest side (2500), Set 3 can form a right triangle.

Set 4: 9 in, 12 in, 15 in

Calculate the sum of the squares of the two shorter sides: 9^2 + 12^2 = 81 + 144 = 225
Calculate the square of the length of the longest side: 15^2 = 225
Compare the two values: 225 = 225

Since the sum of the squares of the two shorter sides (225) is equal to the square of the length of the longest side (225), Set 4 can form a right triangle.

Conclusion

In conclusion, only two of the four sets of side lengths given in the problem can form a right triangle. Set 3: 14 cm, 48 cm, 50 cm and Set 4: 9 in, 12 in, 15 in satisfy the Pythagorean theorem and can form a right triangle. The other two sets, Set 1: 10 cm, 26 cm, 24 cm and Set 2: 4 in, 2 in, 6 in, do not satisfy the theorem and cannot form a right triangle.

Understanding the Importance of the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields, including physics, engineering, and architecture. It is used to determine the length of the hypotenuse of a right triangle, which is essential in calculating distances, heights, and other measurements. The theorem is also used in trigonometry to solve problems involving right triangles.

Real-World Applications of the Pythagorean Theorem

The Pythagorean theorem has numerous real-world applications, including:

Building design: Architects use the theorem to calculate the height of buildings, the length of roofs, and the distance between walls.
Physics: The theorem is used to calculate the distance traveled by an object under the influence of gravity, the height of a projectile, and the distance between two points in space.
Engineering: The theorem is used to calculate the stress and strain on materials, the distance between two points in a mechanical system, and the length of a cable or a rope.
Navigation: The theorem is used to calculate the distance between two points on the surface of the Earth, the height of a mountain, and the distance between two points in space.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that has numerous applications in various fields. It is used to determine the length of the hypotenuse of a right triangle, which is essential in calculating distances, heights, and other measurements. The theorem is also used in trigonometry to solve problems involving right triangles. Only two of the four sets of side lengths given in the problem can form a right triangle, and the theorem has numerous real-world applications in fields such as building design, physics, engineering, and navigation.