Pythagorean Theorem, Distance, and Midpoint

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

Back

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

Front

What is the formula for finding the midpoint of a line segment with endpoints (x1, y1) and (x2, y2)?

Back

The midpoint M of a line segment can be found using the formula: M = ((x1 + x2)/2, (y1 + y2)/2).

Front

If a triangle has sides of lengths 3 and 4, what is the length of the hypotenuse?

Back

Using the Pythagorean Theorem: c = √(3² + 4²) = √(9 + 16) = √25 = 5.

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

Find the midpoint of the segment with endpoints (2, 3) and (4, 7).

Back

Midpoint M = ((2 + 4)/2, (3 + 7)/2) = (3, 5).

Front

What is the length of the diagonal of a rectangle with width 8 inches and height 6 inches?

Back

Using the Pythagorean Theorem: d = √(8² + 6²) = √(64 + 36) = √100 = 10 inches.

Front

If the midpoint of a segment is (5, 5) and one endpoint is (3, 7), what is the other endpoint?

Back

Let the other endpoint be (x, y). Using the midpoint formula: (5, 5) = ((3 + x)/2, (7 + y)/2). Solving gives (7, 3) as the other endpoint.

Front

What is the total distance traveled if a person walks 3 miles east, then 4 miles north?

Back

Using the Pythagorean Theorem: Total distance = √(3² + 4²) = √(9 + 16) = √25 = 5 miles.

Front

How do you find the distance between points (0, 0) and (x, y)?

Back

Using the distance formula: d = √(x² + y²).