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: $$a^2 + b^2 = c^2$$ .

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

The hypotenuse is the longest side of a right triangle, opposite the right angle.

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

The formula is derived from the Pythagorean Theorem: If you know two sides, you can find the third side using $$c = \text{sqrt}(a^2 + b^2)$$ or $$a = \text{sqrt}(c^2 - b^2)$$ .

In a 5-12-13 triangle, it is a right triangle because $$5^2 + 12^2 = 13^2$$ (25 + 144 = 169).

Using the Pythagorean Theorem: $$8^2 + 15^2 = c^2$$ , so $$64 + 225 = c^2$$ , thus $$c = 17$$ .

A Pythagorean triplet consists of three positive integers a, b, and c, such that $$a^2 + b^2 = c^2$$ . Example: (3, 4, 5).

No, the Pythagorean Theorem only applies to right triangles.

Using the Pythagorean Theorem: $$6^2 + 8^2 = c^2$$ , so $$36 + 64 = c^2$$ , thus $$c = 10$$ .