Using the Pythagorean Theorem to Determine if a Triangle is a Right Triangle

Oct 13, 2025 by talynt.org 106 views

Understanding the Concept of a Right Triangle

A right triangle is a triangle with one angle that is equal to 90 degrees. This angle is often referred to as the right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle. The other two sides are called the legs of the triangle.

Using the Pythagorean Theorem to Determine if a Triangle is a Right Triangle

One of the most common methods for determining if a triangle is a right triangle is by using 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 of the triangle, and c is the length of the hypotenuse.

Applying the Pythagorean Theorem to the Given Triangle

Let's apply the Pythagorean theorem to the given triangle with side lengths $5 \, \text{cm}, 13 \, \text{cm},$ and $12 \, \text{cm}$ . We can assume that the side length of 13 cm is the length of the hypotenuse, and the side lengths of 5 cm and 12 cm are the lengths of the legs of the triangle.

If we square the lengths of the legs of the triangle, we get:

5^2 = 25 12^2 = 144

If we add the squares of the lengths of the legs of the triangle, we get:

25 + 144 = 169

Now, let's square the length of the hypotenuse:

13^2 = 169

As we can see, the sum of the squares of the lengths of the legs of the triangle (169) is equal to the square of the length of the hypotenuse (169). This means that the given triangle satisfies the Pythagorean theorem.

Conclusion

Based on the Pythagorean theorem, we can conclude that the triangle with side lengths $5 \, \text{cm}, 13 \, \text{cm},$ and $12 \, \text{cm}$ is a right triangle. The side length of 13 cm is the length of the hypotenuse, and the side lengths of 5 cm and 12 cm are the lengths of the legs of the triangle.

Answer

The correct answer is:

A. The triangle is a right triangle because $5^2 + 12^2 = 13^2$ .

This answer is correct because it accurately states that the triangle is a right triangle based on the Pythagorean theorem.

Discussion

The Pythagorean theorem is a fundamental concept in geometry that is used to determine if a triangle is a right triangle. By applying the Pythagorean theorem to the given triangle, we can conclude that it is a right triangle. This concept is important in mathematics and has many real-world applications, such as in architecture, engineering, and physics.

Real-World Applications

The Pythagorean theorem has many real-world applications, such as:

Architecture: The Pythagorean theorem is used to calculate the height of buildings and the length of shadows.
Engineering: The Pythagorean theorem is used to calculate the stress on bridges and the length of cables.
Physics: The Pythagorean theorem is used to calculate the distance between objects and the length of time it takes for objects to fall.

Conclusion

In conclusion, the Pythagorean theorem is a fundamental concept in geometry that is used to determine if a triangle is a right triangle. By applying the Pythagorean theorem to the given triangle, we can conclude that it is a right triangle. This concept is important in mathematics and has many real-world applications.

Final Answer

The final answer is:

A. The triangle is a right triangle because $5^2 + 12^2 = 13^2$ .

Q: What is a right triangle?

A: A right triangle is a triangle with one angle that is equal to 90 degrees. This angle is often referred to as the right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical formula that is used to determine if a triangle is a right triangle. 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 use the Pythagorean theorem to determine if a triangle is a right triangle?

A: To use the Pythagorean theorem, you need to square the lengths of the legs of the triangle and add them together. If the result is equal to the square of the length of the hypotenuse, then the triangle is a right triangle.

Q: What are the legs of a right triangle?

A: The legs of a right triangle are the two sides that meet at a right angle. They are the sides that are not the hypotenuse.

Q: What is the hypotenuse of a right triangle?

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

Q: Can a triangle have more than one right angle?

A: No, a triangle can only have one right angle. If a triangle has more than one right angle, then it is not a triangle.

Q: Can a right triangle have all sides of equal length?

A: No, a right triangle cannot have all sides of equal length. If a triangle has all sides of equal length, then it is an equilateral triangle, not a right triangle.

Q: Can a right triangle have a right angle of 180 degrees?

A: No, a right triangle cannot have a right angle of 180 degrees. A right angle is defined as an angle that is equal to 90 degrees.

Q: Can a right triangle have a hypotenuse that is shorter than one of the legs?

A: No, a right triangle cannot have a hypotenuse that is shorter than one of the legs. The hypotenuse is always the longest side of the triangle.

Q: Can a right triangle have a hypotenuse that is equal to one of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to one of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the sum of the lengths of the legs?

A: Yes, a right triangle can have a hypotenuse that is equal to the sum of the lengths of the legs. This is a special case of a right triangle called a Pythagorean triple.

Q: What is a Pythagorean triple?

A: A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem:

a^2 + b^2 = c^2

Q: Can a Pythagorean triple have a hypotenuse that is equal to the sum of the lengths of the legs?

A: Yes, a Pythagorean triple can have a hypotenuse that is equal to the sum of the lengths of the legs. This is a special case of a Pythagorean triple.

Q: Can a right triangle have a hypotenuse that is equal to the difference of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the difference of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the product of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the product of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the quotient of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the quotient of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the product of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the product of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs?

A: Yes, a right triangle can have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs. This is a special case of a right triangle called a Pythagorean triple.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the quotient of the squares of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the quotient of the squares of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the product of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the product of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the quotient of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the quotient of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs and the product of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs and the product of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs and the quotient of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs and the quotient of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the product of the squares of the lengths of the legs and the quotient of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the product of the squares of the lengths of the legs and the quotient of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the quotient of the squares of the lengths of the legs and the product of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the quotient of the squares of the lengths of the legs and the product of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the difference of the squares of the lengths of the legs?

A: No, a right triangle cannot have a hypotenuse that is equal to the square root of the sum of the squares of the lengths of the legs and the difference of the squares of the lengths of the legs. The hypotenuse is always longer than the legs.

Q: Can a right triangle have a hypotenuse that is equal to the square root of the difference of the squares of the lengths of the legs and the sum of the squares of the lengths of the legs?

A: No, a right triangle