Pythagorean Theorem

Beginner (high-school level)

Goal of this page

Understand the meaning of the Pythagorean theorem $a^2+b^2=c^2$, its representative proofs, the converse, Pythagorean triples, and the application to the distance formula.

1. The Pythagorean Theorem

The Pythagorean theorem is the most basic relation that holds among the lengths of the $3$ sides of a right triangle.

Figure 1: A right triangle (the angle at $A$ is the right angle). For the hypotenuse $c$ and the other two sides $a,b$, we have $a^2+b^2=c^2$.

The Pythagorean theorem

In a right triangle, let the lengths of the $2$ sides forming the right angle be $a, b$ and the hypotenuse be $c$. Then

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

Example

If $a=3,\ b=4$, then $c=\sqrt{9+16}=\sqrt{25}=5$. The triple $(3,4,5)$ is the most famous right triangle.

2. Proof (by area)

Figure 2: Arranging $4$ congruent right triangles inside a square of side $a+b$ leaves a square of side $c$ at the center.

Proof by arranging triangles inside a square

Inside a square of side $a+b$, arrange $4$ congruent right triangles. A square of side $c$ remains at the center. Expressing the area of the large square in $2$ ways gives

$$(a+b)^2 = 4\cdot\dfrac{1}{2}ab + c^2$$

Expanding the left-hand side, $a^2+2ab+b^2 = 2ab + c^2$. Subtracting $2ab$ from both sides gives $a^2+b^2=c^2$. $\square$

3. The Converse and Applications

The converse

Converse of the Pythagorean theorem

If the side lengths $a, b, c$ of a triangle satisfy $a^2+b^2=c^2$, then the angle opposite the side of length $c$ is a right angle. That is, the triangle is a right triangle.

Proof using congruence

Figure 3: The given triangle $ABC$ (left) and a right triangle $A'B'C'$ (right) whose two sides forming the right angle are $a, b$. The corresponding $3$ sides $a, b, c$ are respectively equal.

Let the $3$ sides of triangle $ABC$ be $a=BC,\ b=CA,\ c=AB$, and suppose $a^2+b^2=c^2$ holds (Figure 3, left).

Separately, construct a new right triangle $A'B'C'$ whose two sides forming the right angle are $b$ and $a$ ($\angle C'=90^\circ,\ C'A'=b,\ C'B'=a$) (Figure 3, right). By the forward Pythagorean theorem, the length of its hypotenuse is

$$A'B' = \sqrt{a^2+b^2} = \sqrt{c^2} = c$$

Then $ABC$ and $A'B'C'$ have their $3$ sides respectively equal ($BC=B'C'=a,\ CA=C'A'=b,\ AB=A'B'=c$), so they are congruent by SSS (three sides equal). In congruent triangles the corresponding angles are also equal, so

$$\angle C = \angle C' = 90^\circ$$

holds. Therefore $ABC$ is a right triangle with the right angle at $\angle C$. $\square$

Alternative proof (law of cosines): Substituting $a^2+b^2=c^2$ into the law of cosines $c^2=a^2+b^2-2ab\cos C$ gives $2ab\cos C=0$. Since $a,b>0$, we get $\cos C=0$, hence $C=90^\circ$ immediately.

Applications

Constructing a right angle: ancient Egyptian rope-stretchers tied equally spaced knots into a rope to form a $(3,4,5)$ triangle and laid out right angles on the ground. This is an old, practical use of the converse.
Pythagorean triples: sets of natural numbers satisfying $a^2+b^2=c^2$, such as $(3,4,5),(5,12,13),(8,15,17)$.
Distance formula: the distance between two points $(x_1,y_1),(x_2,y_2)$ in the coordinate plane is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. This is the Pythagorean theorem applied to coordinates.
Relation to the law of cosines: the law of cosines extends the Pythagorean theorem to an arbitrary angle; the case of a right angle is exactly the Pythagorean theorem.

FAQ

Q1. What is the Pythagorean theorem?

In a right triangle, if the two sides forming the right angle are a and b and the hypotenuse is c, then a²+b²=c². The most famous example is the (3,4,5) right triangle.

Q2. Does the converse hold?

Yes. Any triangle whose three sides satisfy a²+b²=c² is necessarily a right triangle (the converse). In antiquity a rope with equally spaced knots was tied into a (3,4,5) triangle to lay out a right angle.

Q3. What is a Pythagorean triple?

A set of natural numbers satisfying a²+b²=c². Examples include (3,4,5), (5,12,13), and (8,15,17); when they are coprime they are called a primitive Pythagorean triple.

Related pages on this site

Law of Cosines (Geometry, Introduction) — extension to an arbitrary angle
Heron's Formula (Geometry, Introduction) — the area of a triangle
The Pythagorean Trigonometric Identity (Geometry, Introduction) — relation to $\sin^2\theta+\cos^2\theta=1$

References

Maor, E. (2007). The Pythagorean Theorem: A 4,000-Year History, Princeton University Press.
Wikipedia: Pythagorean theorem