4.2: Pythagorean Theorem & 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

How can you determine if a triangle is a right triangle using the Pythagorean Theorem?

Back

To determine if a triangle is a right triangle, check if the lengths of the sides satisfy the Pythagorean Theorem: a² + b² = c², where c is the longest side.

Front

What are the side lengths of a right triangle?

Back

In a right triangle, the side lengths can be any positive numbers that satisfy the Pythagorean Theorem. Common examples include (3, 4, 5) and (5, 12, 13).

Front

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.

Front

If a triangle has sides of lengths 7, 24, and 25, how can you verify if it is a right triangle?

Back

To verify, apply the Pythagorean Theorem: 7² + 24² = 49 + 576 = 625, and 25² = 625. Since both sides are equal, it is a right triangle.

Front

What is the formula to find the length of the diagonal of a rectangle?

Back

The length of the diagonal (d) of a rectangle can be found using the formula: d = √(length² + width²).

Front

How do you find the height of a right triangle if the base and hypotenuse are known?

Back

Use the Pythagorean Theorem: h = √(c² - a²), where c is the hypotenuse and a is the base.

Front

What is the significance of the longest side in a right triangle?

Back

The longest side in a right triangle is called the hypotenuse, and it is opposite the right angle. It is always the side used in the Pythagorean Theorem.

Front

Can a triangle with sides 2.1, 7.2, and 7.5 be a right triangle?

Back

Yes, to check, calculate: 2.1² + 7.2² = 4.41 + 51.84 = 56.25 and 7.5² = 56.25. Since both sides are equal, it is a right triangle.

Front

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

Back

In a right triangle, the side opposite the right angle is the longest (hypotenuse), and the other two sides are called the legs. The angles opposite the legs are acute.