Applying the Pythagorean Theorem to the Given Sets of Side Lengths

Oct 13, 2025 by pynkocean.com 133 views

Understanding the Basics of Right Triangles

A right triangle is a triangle with one angle that is 90 degrees. This type of triangle is also known as a right-angled triangle. 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 helps us determine whether a triangle is a right triangle or not.

The Pythagorean Theorem

The Pythagorean theorem states that in a right 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 expressed mathematically as:

a^2 + b^2 = c^2

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Given Sets of Side Lengths

We are given four sets of side lengths, and we need to determine which ones will form a right triangle. Let's apply the Pythagorean theorem to each set:

Set 1: 10 cm, 26 cm, 24 cm

To determine whether this set forms a right triangle, we need to check if the square of the longest side (26 cm) is equal to the sum of the squares of the other two sides (10 cm and 24 cm).

26^2 = 676 10^2 + 24^2 = 100 + 576 = 676

Since 26^2 = 10^2 + 24^2, this set forms a right triangle.

Set 2: 4 in., 2 in., 6 in.

To determine whether this set forms a right triangle, we need to check if the square of the longest side (6 in.) is equal to the sum of the squares of the other two sides (4 in. and 2 in.).

6^2 = 36 4^2 + 2^2 = 16 + 4 = 20

Since 6^2 ≠ 4^2 + 2^2, this set does not form a right triangle.

Set 3: 15 cm, 20 cm, 25 cm

To determine whether this set forms a right triangle, we need to check if the square of the longest side (25 cm) is equal to the sum of the squares of the other two sides (15 cm and 20 cm).

25^2 = 625 15^2 + 20^2 = 225 + 400 = 625

Since 25^2 = 15^2 + 20^2, this set forms a right triangle.

Set 4: 3 in., 4 in., 5 in.

To determine whether this set forms a right triangle, we need to check if the square of the longest side (5 in.) is equal to the sum of the squares of the other two sides (3 in. and 4 in.).

5^2 = 25 3^2 + 4^2 = 9 + 16 = 25

Since 5^2 = 3^2 + 4^2, this set forms a right triangle.

Conclusion

In conclusion, the sets of side lengths that will form a right triangle are:

Set 1: 10 cm, 26 cm, 24 cm
Set 3: 15 cm, 20 cm, 25 cm
Set 4: 3 in., 4 in., 5 in.

These sets satisfy the Pythagorean theorem, which is a necessary and sufficient condition for a triangle to be a right triangle.

Q: What is a right triangle?

A: A right triangle is a triangle with one angle that is 90 degrees. This type of triangle is also known as a right-angled triangle.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that helps us determine whether a triangle is a right triangle or not. It states that in a right 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 expressed mathematically as:

a^2 + b^2 = c^2

Q: How do I determine if a triangle is a right triangle?

A: To determine if a triangle is a right triangle, you can use the Pythagorean theorem. Simply square the lengths of the three sides, and if the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Q: What is the hypotenuse of a right triangle?

A: The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of the triangle.

Q: Can a right triangle have two right angles?

A: No, a right triangle cannot have two right angles. By definition, a right triangle has one right angle, and the other two angles must be acute angles.

Q: Can a right triangle have an obtuse angle?

A: No, a right triangle cannot have an obtuse angle. The sum of the interior angles of a triangle is always 180 degrees, and since one angle is 90 degrees, the other two angles must be acute angles.

Q: What is the difference between a right triangle and an isosceles triangle?

A: A right triangle is a triangle with one right angle, while an isosceles triangle is a triangle with two sides of equal length. These two types of triangles are not mutually exclusive, and it is possible for a triangle to be both a right triangle and an isosceles triangle.

Q: Can a right triangle be equilateral?

A: No, a right triangle cannot be equilateral. An equilateral triangle has all three sides of equal length, while a right triangle has one right angle and two sides of different lengths.

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

A: To find the length of the hypotenuse of a right triangle, you can use the Pythagorean theorem. Simply square the lengths of the other two sides, add them together, and take the square root of the result.

Q: Can I use the Pythagorean theorem to find the length of a side of a right triangle if I know the lengths of the other two sides?

A: Yes, you can use the Pythagorean theorem to find the length of a side of a right triangle if you know the lengths of the other two sides. Simply square the lengths of the known sides, add them together, and take the square root of the result.

Q: What are some real-world applications of right triangles?

A: Right triangles have many real-world applications, including:

Building design and construction
Physics and engineering
Navigation and surveying
Computer graphics and game development
Architecture and design

Q: Can I use a calculator to find the length of the hypotenuse of a right triangle?

A: Yes, you can use a calculator to find the length of the hypotenuse of a right triangle. Simply enter the lengths of the other two sides, and the calculator will give you the length of the hypotenuse.

Q: Can I use a computer program to find the length of the hypotenuse of a right triangle?

A: Yes, you can use a computer program to find the length of the hypotenuse of a right triangle. Many computer programs, including calculators and geometry software, can perform this calculation for you.