Understanding the Pythagorean Theorem

What is the Pythagorean Theorem?

Answer explanation

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). Thus, the correct formula is a² + b² = c².

In a right triangle, if one leg is 3 units and the other leg is 4 units, what is the length of the hypotenuse?

• 5 units

• 6 units

• 7 units

• 8 units

Answer explanation

In a right triangle, use the Pythagorean theorem: a² + b² = c². Here, 3² + 4² = 9 + 16 = 25. Thus, c = √25 = 5. Therefore, the hypotenuse is 5 units.

• The shortest side

• The hypotenuse

• The base

• The height

Answer explanation

In a right triangle, the Pythagorean Theorem refers to the hypotenuse as 'c'. The hypotenuse is the longest side, opposite the right angle, while the other two sides are shorter.

If the hypotenuse of a right triangle is 13 units and one leg is 5 units, what is the length of the other leg?

• 8 units

• 10 units

• 12 units

• 9 units

Answer explanation

Using the Pythagorean theorem, a² + b² = c², where c is the hypotenuse. Here, 5² + b² = 13². This simplifies to 25 + b² = 169. Thus, b² = 144, and b = 12. Therefore, the length of the other leg is 12 units.

Which of the following sets of numbers can be the lengths of the sides of a right triangle?

• 2, 3, 4

• 5, 12, 13

• 6, 8, 10

• Both B and C

Answer explanation

To determine if a set of numbers can form a right triangle, we use the Pythagorean theorem: a² + b² = c². For 5, 12, 13: 5² + 12² = 25 + 144 = 169 = 13². For 6, 8, 10: 6² + 8² = 36 + 64 = 100 = 10². Thus, both B and C are correct.

• 6

• 8

• 12

• 4

Answer explanation

In a right triangle, we can use the Pythagorean theorem: a² + b² = c². Here, a = 6 and c = 10. Thus, 6² + b² = 10², which simplifies to 36 + b² = 100. Solving for b² gives b² = 64, so b = 8.

Which of the following is NOT a correct application of the Pythagorean Theorem?

• Finding the length of the hypotenuse in a right triangle

• Finding the area of a rectangle

• Checking if a triangle is a right triangle

• Finding the length of a missing leg in a right triangle

Answer explanation

The Pythagorean Theorem applies to right triangles, not rectangles. It is used to find the hypotenuse or a leg of a right triangle and to check if a triangle is right, but not for calculating the area of a rectangle.

If the legs of a right triangle are both 7 units, what is the length of the hypotenuse?

Answer explanation

In a right triangle, the hypotenuse can be found using the Pythagorean theorem: c = √(a² + b²). Here, a = 7 and b = 7, so c = √(7² + 7²) = √(49 + 49) = √98 = 7√2. Thus, the hypotenuse is 7√2.

A ladder is leaning against a wall. The foot of the ladder is 9 feet from the wall, and the ladder reaches 12 feet up the wall. How long is the ladder?

• 15 feet

• 13 feet

• 21 feet

• 10 feet

Answer explanation

To find the length of the ladder, use the Pythagorean theorem: a² + b² = c². Here, a = 9 feet (distance from the wall), b = 12 feet (height up the wall). Thus, 9² + 12² = c², or 81 + 144 = c², giving c = 15 feet.

• 17

• 16

• 13

• 12

Answer explanation

In a right triangle, the Pythagorean theorem states that $c^2 = a^2 + b^2$. Here, $c^2 = 8^2 + 15^2 = 64 + 225 = 289$. Thus, $c = \sqrt{289} = 17$. Therefore, the value of $c$ is 17.