Pythagorean Theorem

Pythagorean Theorem

How to solve the Pythagorean theorem problems: theorem, proof, examples, and their solutions.

Theorem

If the lengths of the right triangle's legs are a and b, and if the length of the right triangle's hypotenuse is c, then a^2 + b^2 = c^2. This is the Pythagorean theorem (or the Pythagoras' theorem).

For a right triangle,
if legs: a, b and hypotenuse: c,
then a2 + b2 = c2.

This is the Pythagorean theorem.
(= Pythagoras' theorem)

Proof

Start from the right triangle whose lengths of the legs are a and b, and whose length of the hypotenuse is c. Draw two square using four of the right triangles. Then (a + b)^2 = 4*(1/2)ab + c^2. Then this becomes a^2 + b^2 = c^2.

Start from the right triangle
whose lengths of the legs are a and b,
and whose length of the hypotenuse is c.

Draw two squares using four of the right triangles.

The center quadrilateral's angle is a right angle,
because m∠(plain) + m∠(dot) + 90 = 180.
(See the right triangle's interior angles.)

Find the areas of both squares:

(purple square) = (a + b)2

(right square) = 4 × (1/2)ab + c2
= (4 × blue triangle) + (red square)

The areas of both squares are equal:
(a + b)2 = 4 × (1/2)ab + c2

(a + b)2 = a2 + 2ab + b2

Square of a sum

Example 1

Find the value of x. The lengths of the right triangle's legs: 3, 4. The length of the right triangle's hypotenuse: x.

Legs: 3, 4
Hypotenuse: x

Example 2

Find the value of x. The lengths of the right triangle's legs: 5, x. The length of the right triangle's hypotenuse: square root 89.

Legs: 5, x
Hypotenuse: √89