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

Using the Pythagorean Theorem: $$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$ , so $$c = \text{sqrt}(169) = 13$$ .

The Pythagorean Theorem only applies to right triangles.

The hypotenuse is 17, as it is the longest side.

True.

The formula is $$6^2 + b^2 = 10^2$$ .

Check: $$9^2 + 12^2 = 15^2$$ . This simplifies to $$81 + 144 = 225$$ , which is true.

In a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side (hypotenuse).

Use the Pythagorean Theorem: $$5^2 + b^2 = 13^2$$ , which simplifies to $$25 + b^2 = 169$$ , so $$b^2 = 144$$ , thus $$b = 12$$ .