Pythagorean Theorem and Special Right Triangles

Front

What is the Pythagorean Theorem?

Back

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). It can be expressed as: c² = a² + b².

Front

What are the special right triangles?

Back

The special right triangles are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the legs are equal, and the hypotenuse is √2 times the length of a leg. In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2.

Front

How do you find the length of the hypotenuse in a right triangle?

Back

To find the length of the hypotenuse (c) in a right triangle, use the Pythagorean Theorem: c = √(a² + b²), where a and b are the lengths of the other two sides.

Front

What is the length of the legs in a 45-45-90 triangle if the hypotenuse is 10?

Back

In a 45-45-90 triangle, the legs are equal and can be found using the formula: leg = hypotenuse / √2. Therefore, if the hypotenuse is 10, each leg is 10/√2 = 5√2.

Front

What is the ratio of the sides in a 30-60-90 triangle?

Back

The ratio of the sides in a 30-60-90 triangle is 1:√3:2, where the side opposite the 30° angle is 1, the side opposite the 60° angle is √3, and the hypotenuse is 2.

Front

If one leg of a 30-60-90 triangle is 5, what is the length of the hypotenuse?

Back

In a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg. Therefore, if one leg is 5, the hypotenuse is 5 * 2 = 10.

Front

What is the value of x in a right triangle with legs of length 3 and 4?

Back

Using the Pythagorean Theorem: c² = 3² + 4² = 9 + 16 = 25. Therefore, x (the hypotenuse) = √25 = 5.

Front

How do you determine if a set of three lengths can form a right triangle?

Back

To determine if three lengths can form a right triangle, check if the square of the longest length equals the sum of the squares of the other two lengths (Pythagorean Theorem).

Front

What is the value of y if one leg of a right triangle is 6 and the hypotenuse is 10?

Back

Using the Pythagorean Theorem: 10² = 6² + y². Therefore, 100 = 36 + y², which gives y² = 64, so y = 8.

Front

What is the area of a right triangle with legs of length 5 and 12?

Back

The area of a right triangle is given by the formula: Area = (1/2) * base * height. Therefore, Area = (1/2) * 5 * 12 = 30.