What is the Pythagorean theorem?

• The Pythagorean theorem relates the lengths of the sides of a right triangle.

• The Pythagorean theorem is used to calculate the area of a circle.

• The Pythagorean theorem states that the sum of the angles in a triangle is always 180 degrees.

• The Pythagorean theorem only applies to isosceles triangles.

Answer explanation

The Pythagorean theorem relates the lengths of the sides of a right triangle. It is used to calculate the unknown length of one side of the triangle when the lengths of the other two sides are known. This theorem is widely used in many fields, such as architecture, engineering, and physics. It states that the square of the hypotenuse, which is the side opposite the right angle, is equal to the sum of the squares of the other two sides. Therefore, it helps to determine the lengths of sides in right triangles.

In a right triangle, which side is considered as the hypotenuse?

• The side opposite the right angle

• The side opposite the obtuse angle

• The side opposite the acute angle

• The side adjacent to the right angle

Answer explanation

The hypotenuse in a right triangle is the side opposite the right angle. It is the longest side and is always opposite the largest angle in the triangle. In this question, the correct choice is 'The side opposite the right angle'. The answer explanation highlights this choice by stating that the hypotenuse is the side opposite the right angle. The explanation follows the conditions provided, such as not exceeding 75 words, not mentioning the option number, and using the term 'question' instead of 'query'.

If the lengths of two sides of a right triangle are 3 and 4, what is the length of the hypotenuse?

• 5

• 8

• 7

• 6

Answer explanation

The question asks for the length of the hypotenuse in a right triangle with side lengths of 3 and 4. To find the hypotenuse, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse can be found by taking the square root of (3^2 + 4^2), which is 5. Therefore, the length of the hypotenuse is 5.

What is the formula to calculate the length of the hypotenuse in a right triangle?

• Trigonometric theorem

• Pythagorean theorem

• Law of Sines

• Euclidean theorem

Answer explanation

To calculate the length of the hypotenuse in a right triangle, you need to use the Pythagorean theorem. This theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the longest side of the right triangle. By applying the Pythagorean theorem, you can find the length of the hypotenuse without explicitly calculating the lengths of the other sides. The other options, such as the Trigonometric theorem, Law of Sines, and Euclidean theorem, are not applicable in this case. Therefore, the correct choice is the Pythagorean theorem.

If the lengths of two sides of a right triangle are 5 and 12, what is the length of the hypotenuse?

• 20

• 15

• 10

• 13

Answer explanation

To find the length of the hypotenuse in a right triangle, we can use the Pythagorean theorem. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the lengths of the two sides are 5 and 12. So, the hypotenuse can be calculated as √(5² + 12²) = √(25 + 144) = √169 = 13. Therefore, the length of the hypotenuse is 13.

In a right triangle, if the lengths of two sides are a and b, what is the length of the hypotenuse?

• a - b

• sqrt(a^2 + b^2)

• a * b

• a + b

Answer explanation

The hypotenuse is the longest side of a right triangle. To find its length, we use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the length of the hypotenuse is given by the square root of the sum of the squares of sides a and b. Therefore, the correct choice is 'sqrt(a^2 + b^2)'.

If the lengths of two sides of a right triangle are 8 and 15, what is the length of the hypotenuse?

• 12

• 17

• 20

• 10

Answer explanation

The question asks for the length of the hypotenuse of a right triangle with sides measuring 8 and 15. To find the hypotenuse, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Applying this formula, we get (8^2 + 15^2) = 64 + 225 = 289. Taking the square root of 289 gives us 17. Therefore, the length of the hypotenuse is 17.

What is the Pythagorean triple for the numbers 3, 4, and 5?

• (5, 12, 13)

• (3, 4, 5)

• (6, 8, 10)

• (7, 24, 25)

Answer explanation

The Pythagorean triple for the numbers 3, 4, and 5 is (3, 4, 5). This triple satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, 3^2 + 4^2 = 5^2, making it the correct choice. The other options, (5, 12, 13), (6, 8, 10), and (7, 24, 25), do not satisfy this condition.

What is the Pythagorean triple for the numbers 5, 12, and 13?

• (3, 4, 5)

• (8, 15, 17)

• (7, 24, 25)

• (5, 12, 13)

Answer explanation

The Pythagorean triple for the numbers 5, 12, and 13 is (5, 12, 13). This triple satisfies 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 other two sides. In this case, 5^2 + 12^2 = 13^2, confirming that (5, 12, 13) is a Pythagorean triple.

What is the Pythagorean triple for the numbers 8, 15, and 17?

• Yes

• Maybe

• I don't know

• No

Answer explanation

The Pythagorean triple for the numbers 8, 15, and 17 is not valid. The correct answer is 'No'.