Frequently Asked Questions About Right Triangles and the Pythagorean Theorem

Oct 13, 2025 by pynkocean.com 126 views

Which of the Following Sets Could be the Sides of a Right Triangle?

In mathematics, a right triangle is a triangle in which one of the angles is a right angle (90 degrees). The sides of a right triangle are related by the Pythagorean theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this article, we will explore which sets of numbers could be the sides of a right triangle.

The Pythagorean theorem is a fundamental concept in geometry and trigonometry. It states that in a right triangle with legs of length a and b, and a hypotenuse of length c, the following equation holds:

a^2 + b^2 = c^2

This theorem can be used to determine whether a set of numbers could be the sides of a right triangle. We will use this theorem to analyze the sets given in the problem.

Let's analyze the first set: {2, 3, √13}. To determine whether this set could be the sides of a right triangle, we need to check if the Pythagorean theorem holds. We can do this by plugging in the values of a and b into the equation:

a^2 + b^2 = c^2

In this case, a = 2, b = 3, and c = √13. Plugging these values into the equation, we get:

2^2 + 3^2 = (√13)^2 4 + 9 = 13 13 = 13

Since the equation holds, this set could be the sides of a right triangle.

Now, let's analyze the second set: (1, 3, √3). To determine whether this set could be the sides of a right triangle, we need to check if the Pythagorean theorem holds. We can do this by plugging in the values of a and b into the equation:

a^2 + b^2 = c^2

In this case, a = 1, b = 3, and c = √3. Plugging these values into the equation, we get:

1^2 + 3^2 = (√3)^2 1 + 9 = 3 10 ≠ 3

Since the equation does not hold, this set could not be the sides of a right triangle.

Finally, let's analyze the third set: {2, 2, 4}. To determine whether this set could be the sides of a right triangle, we need to check if the Pythagorean theorem holds. We can do this by plugging in the values of a and b into the equation:

a^2 + b^2 = c^2

In this case, a = 2, b = 2, and c = 4. Plugging these values into the equation, we get:

2^2 + 2^2 = 4^2 4 + 4 = 16 8 ≠ 16

Since the equation does not hold, this set could not be the sides of a right triangle.

In conclusion, only set A: {2, 3, √13} could be the sides of a right triangle. The Pythagorean theorem holds for this set, while it does not hold for sets B and C. This demonstrates the importance of the Pythagorean theorem in determining whether a set of numbers could be the sides of a right triangle.

Here are a few more examples of sets that could be the sides of a right triangle:

{3, 4, 5}
{5, 12, 13}
{8, 15, 17}

These sets all satisfy the Pythagorean theorem, and therefore could be the sides of a right triangle.

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

Building design: Architects use the Pythagorean theorem to determine the length of the hypotenuse of a right triangle, which is necessary for designing buildings and other structures.
Surveying: Surveyors use the Pythagorean theorem to determine the distance between two points, which is necessary for mapping out land and other territories.
Physics: Physicists use the Pythagorean theorem to calculate the distance and velocity of objects in motion.

In conclusion, the Pythagorean theorem is a fundamental concept in geometry and trigonometry that has many real-world applications. By understanding the Pythagorean theorem, we can determine whether a set of numbers could be the sides of a right triangle, and apply this knowledge to a variety of fields.
Frequently Asked Questions About Right Triangles and the Pythagorean Theorem

A: The Pythagorean theorem is a fundamental concept in geometry and trigonometry that states that in a right triangle with legs of length a and b, and a hypotenuse of length c, the following equation holds:

a^2 + b^2 = c^2

A: A right triangle is a triangle in which one of the angles is a right angle (90 degrees). The sides of a right triangle are related by the Pythagorean theorem.

A: To determine whether a set of numbers could be the sides of a right triangle, you need to check if the Pythagorean theorem holds. You can do this by plugging in the values of a and b into the equation:

a^2 + b^2 = c^2

If the equation holds, then the set of numbers could be the sides of a right triangle.

A: Some common mistakes to avoid when using the Pythagorean theorem include:

Rounding errors: Make sure to use exact values when plugging in the values of a and b into the equation.
Sign errors: Make sure to use the correct signs for the values of a and b.
Order of operations: Make sure to follow the order of operations when plugging in the values of a and b into the equation.

A: Yes, the Pythagorean theorem can be used to solve problems in other areas of mathematics, including:

Trigonometry: The Pythagorean theorem can be used to solve problems involving right triangles and trigonometric functions.
Geometry: The Pythagorean theorem can be used to solve problems involving right triangles and geometric shapes.
Physics: The Pythagorean theorem can be used to solve problems involving motion and velocity.

A: The Pythagorean theorem can be applied in a variety of real-world situations, including:

Building design: Architects use the Pythagorean theorem to determine the length of the hypotenuse of a right triangle, which is necessary for designing buildings and other structures.
Surveying: Surveyors use the Pythagorean theorem to determine the distance between two points, which is necessary for mapping out land and other territories.
Physics: Physicists use the Pythagorean theorem to calculate the distance and velocity of objects in motion.

A: Some common applications of the Pythagorean theorem include:

Distance and velocity calculations: The Pythagorean theorem can be used to calculate the distance and velocity of objects in motion.
Building design: The Pythagorean theorem can be used to determine the length of the hypotenuse of a right triangle, which is necessary for designing buildings and other structures.
Surveying: The Pythagorean theorem can be used to determine the distance between two points, which is necessary for mapping out land and other territories.

A: No, the Pythagorean theorem can only be used to solve problems involving right triangles. However, there are other theorems and formulas that can be used to solve problems involving non-right triangles.

A: Some common misconceptions about the Pythagorean theorem include:

The Pythagorean theorem only applies to right triangles: The Pythagorean theorem can only be used to solve problems involving right triangles.
The Pythagorean theorem is only used in mathematics: The Pythagorean theorem has many real-world applications and can be used in a variety of fields, including physics, engineering, and architecture.