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

A right triangle is a triangle that has one angle measuring 90 degrees.

The sides of a right triangle are called the legs (the two shorter sides) and the hypotenuse (the longest side opposite the right angle).

To determine if a set of lengths can form a right triangle, check if the lengths satisfy the Pythagorean Theorem: $$a^2 + b^2 = c^2$$ , where c is the longest side.

Using the Pythagorean Theorem: $$4^2 + 10^2 = c^2$$ , we find that $$c = \sqrt{116} \approx 10.77$$ .

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

A 3-4-5 triangle is a special right triangle where the lengths of the sides are in the ratio 3:4:5, satisfying the Pythagorean Theorem: $$3^2 + 4^2 = 5^2$$ .

Using the Pythagorean Theorem: $$6^2 + 8^2 = c^2$$ , we find that $$c = 10$$ .

The formula to find the missing side of a right triangle is: $$a^2 + b^2 = c^2$$ , where c is the hypotenuse.

Both sets can form right triangles as they satisfy the Pythagorean Theorem: $$5^2 + 12^2 = 13^2$$ and $$7^2 + 24^2 = 25^2$$ .