8. 1 Pythagorean Theorem & Its Converse

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: a² + b² = c².

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).

Front

How do you determine if a set of numbers is a Pythagorean Triple?

Back

To determine if a set of numbers is a Pythagorean Triple, check if the sum of the squares of the two smaller numbers equals the square of the largest number.

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What is the converse of the Pythagorean Theorem?

Back

The converse of the Pythagorean Theorem states that if a triangle has side lengths a, b, and c, and a² + b² = c², then the triangle is a right triangle.

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What type of triangle is formed if a² + b² > c²?

Back

If a² + b² > c², the triangle is an acute triangle.

Front

What type of triangle is formed if a² + b² < c²?

Back

If a² + b² < c², the triangle is an obtuse triangle.

Front

If one side of a right triangle is 6 in and the other is 8 in, what is the length of the hypotenuse?

Back

Using the Pythagorean Theorem: c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 in.

Front

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

Back

To find the length of a missing side in a right triangle, use the Pythagorean Theorem: a² + b² = c², where c is the hypotenuse.

Front

What is the relationship between the sides of a right triangle?

Back

In a right triangle, the length of the hypotenuse is always greater than the lengths of either of the other two sides.

Front

If a ladder is placed 10 feet away from a wall and reaches a height of 25 feet, how long is the ladder?

Back

Using the Pythagorean Theorem: c² = 10² + 25² = 100 + 625 = 725, so c = √725 ≈ 26.93 feet.