Determining Right Triangles Using the Pythagorean Theorem

Understanding how to determine if a set of numbers can form a right triangle is an important skill in geometry. The Pythagorean Theorem is a fundamental concept in this area: for any right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. This article will walk through the process of determining which sets of numbers could be the lengths of the sides of a right triangle using the Pythagorean Theorem.

We will evaluate the following sets of numbers: (A. 22, 24, 28) (B. 12, 13, 15) (C. 8, 12, 15) (D. 6, 8, 10).

Evaluating Each Set of Numbers

Using the Pythagorean Theorem, which states that for a right triangle with sides a, b, and hypotenuse c (where c is the longest side), the following relationship holds:

a2 b2 c2

Set A: 22, 24, 28

Let's check:

222 242 484 576 1060

282 784

1060 ≠ 784

This is not a right triangle.

Set B: 12, 13, 15

Let's check:

122 132 144 169 313

152 225

313 ≠ 225

This is not a right triangle.

Set C: 8, 12, 15

Let's check:

82 122 64 144 208

152 225

208 ≠ 225

This is not a right triangle.

Set D: 6, 8, 10

Let's check:

62 82 36 64 100

102 100

100 100

This is a right triangle.

Conclusion

The only set of numbers that can be the lengths of the sides of a right triangle is D. 6, 8, 10.

Key Takeaways

In a right triangle: The square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of the triangle. Verification: A set of three numbers can form a right triangle if the square of the largest number is equal to the sum of the squares of the other two numbers.

It is also interesting to note that a Pythagorean triple can never be made up of all odd numbers or two even numbers and one odd number. This explains why we can quickly eliminate C (8, 12, 15) without detailed calculation.

Answer: D (6, 8, 10) is the correct set of numbers that can form a right triangle.