Pythagorean Theorem

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 is expressed as: $$c^2 = a^2 + b^2$$ .

The Pythagorean Theorem only applies to right triangles.

Using the Pythagorean Theorem: $$c^2 = 3^2 + 4^2 = 9 + 16 = 25$$ , so $$c = 5$$ .

To find the length of one side, rearrange the Pythagorean Theorem: For side a, use $$a^2 = c^2 - b^2$$ ; for side b, use $$b^2 = c^2 - a^2$$ .

Use the formula: $$b^2 = c^2 - a^2$$ . Here, $$b^2 = 10^2 - 6^2 = 100 - 36 = 64$$ , so $$b = 8$$ .

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1 : $$\sqrt{3}$$ : 2.

In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is $$\sqrt{2}$$ times the length of one leg.

The Pythagorean Theorem can be used in various real-life situations, such as determining the height of a ladder against a wall or finding the shortest distance between two points.

The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by: $$d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$$ .

Using the Pythagorean Theorem: $$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$ , so $$c = 13$$ .