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 side lengths of a 45-45-90 triangle?

Back

In a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse is √2 times the length of a leg. If each leg is 'x', then the hypotenuse is x√2.

Front

What are the side lengths of a 30-60-90 triangle?

Back

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. If the shortest side (opposite the 30° angle) is 'x', then the longer leg (opposite the 60° angle) is x√3, and the hypotenuse is 2x.

Front

How do you determine if 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 (c² = a² + b²).

Front

If one leg of a right triangle is 5 and the other leg is 12, what is the length of the hypotenuse?

Back

Using the Pythagorean Theorem: c² = 5² + 12²; c² = 25 + 144; c² = 169; c = 13.

Front

What is the relationship between the angles and sides in a right triangle?

Back

In a right triangle, the side opposite the largest angle (90°) is the hypotenuse, and it is always the longest side. The other two sides are called legs.

Front

What is the formula to find the area of a right triangle?

Back

The area of a right triangle can be found using the formula: Area = (1/2) × base × height.

Front

What is the significance of the 5-12-13 triangle?

Back

The 5-12-13 triangle is a Pythagorean triple, meaning it satisfies the Pythagorean theorem (5² + 12² = 13²) and represents a right triangle.

Front

If a ladder is 13 feet long and is placed 5 feet away from a wall, how high does it reach?

Back

Using the Pythagorean theorem: c² = a² + b²; 13² = h² + 5²; 169 = h² + 25; h² = 144; h = 12 feet.

Front

What is a Pythagorean triple?

Back

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Examples include (3, 4, 5) and (5, 12, 13).