Pythagorean Theorem and distance between two points

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 is a Pythagorean Triple?

Back

A Pythagorean Triple consists of three positive integers (a, b, c) that satisfy the Pythagorean Theorem, meaning a² + b² = c². Example: (3, 4, 5) is a Pythagorean Triple.

Front

How do you find the distance between two points (x1, y1) and (x2, y2)?

Back

The distance d can be found using the distance formula: d = √((x2 - x1)² + (y2 - y1)²).

Front

What is the distance between the points (1, 2) and (4, 6)?

Back

Using the distance formula: d = √((4 - 1)² + (6 - 2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Front

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

Back

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

Front

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

Back

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, as stated by the Pythagorean Theorem.

Front

Can the numbers 5, 12, and 13 form a Pythagorean Triple?

Back

Yes, because 5² + 12² = 25 + 144 = 169, and 13² = 169.

Front

What is the area of a square with a side length of 4?

Back

The area A of a square is calculated as A = side². So, A = 4² = 16.

Front

What is the formula for the area of a right triangle?

Back

The area A of a right triangle is given by A = (1/2) × base × height.

Front

If the legs of a right triangle are 9 and 12, what is the length of the hypotenuse?

Back

Using the Pythagorean Theorem: c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15.