Pythagoras theorem and its converse

The most famous theorem in elementary geometry is so important that an entire culture is associated with it. Pythagoras gives us a quick check for whether a triangle is right-angled, and a way to compute the hypotenuse from the two legs.

Statement

Pythagoras Theorem. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

If $\triangle ABC$ has $\angle B = 90^\circ$, with legs $a = BC$ and $c = AB$ and hypotenuse $b = AC$, then $b^2 = a^2 + c^2.$

(Different textbooks use different letters; the result is the same.)

Proof using similarity

Setup. In $\triangle ABC$ with $\angle B = 90^\circ$, drop the altitude $BD$ from $B$ to the hypotenuse $AC$, hitting $AC$ at $D$.

Step 1 , small similar triangles. As shown in the earlier topic on similarity criteria, $\triangle ABD \sim \triangle ABC$ and $\triangle CBD \sim \triangle ABC$ (both by AA, sharing one angle and a right angle).

Step 2 , write the ratios.

From $\triangle ABD \sim \triangle ABC$: $\dfrac{AB}{AC} = \dfrac{AD}{AB} \Rightarrow AB^2 = AD \cdot AC$.

From $\triangle CBD \sim \triangle ABC$: $\dfrac{CB}{AC} = \dfrac{DC}{CB} \Rightarrow CB^2 = DC \cdot AC$.

Step 3 , add. $AB^2 + CB^2 = AD \cdot AC + DC \cdot AC = (AD + DC) \cdot AC = AC \cdot AC = AC^2. \qquad \blacksquare$

This is one of the shortest, most elegant proofs of Pythagoras , and shows similarity is the deeper underlying tool.

Converse

Converse of Pythagoras. If in a triangle the square of one side equals the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

That is, if $a^2 + c^2 = b^2$ for sides $a, b, c$, then the angle opposite $b$ is $90^\circ$.

This is wonderfully useful: it lets us decide whether three given lengths form a right triangle.

Worked examples

Example 1. Find the hypotenuse of a right triangle with legs $5$ cm and $12$ cm.

$h^2 = 25 + 144 = 169 \Rightarrow h = 13$ cm.

Example 2. A $5$-m ladder leans against a wall. Its foot is $3$ m from the wall. How high up the wall does the ladder reach?

The ladder, the wall, and the ground form a right triangle with hypotenuse $5$ and one leg $3$. The other leg satisfies $h^2 + 9 = 25 \Rightarrow h = 4$ m.

Example 3. Sides of a triangle are $7, 24, 25$ cm. Is it a right triangle?

$7^2 + 24^2 = 49 + 576 = 625 = 25^2$. By the converse of Pythagoras, yes , the angle opposite the side of length $25$ is $90^\circ$.

Example 4. In $\triangle ABC$ with $\angle B = 90^\circ$, $BD \perp AC$ at $D$. Show that $BD^2 = AD \cdot DC$.

From $\triangle ABD \sim \triangle CBD$ (each similar to $\triangle ABC$, hence to each other): $\frac{AD}{BD} = \frac{BD}{DC} \Rightarrow BD^2 = AD \cdot DC. \quad \blacksquare$

(This is sometimes called the geometric mean relation in a right triangle.)

Example 5. A right triangle has legs $9$ and $40$. Find the hypotenuse and the area.

$h^2 = 81 + 1600 = 1681 \Rightarrow h = 41$.

Area $= \frac{1}{2} \cdot 9 \cdot 40 = 180$ square units.

Try it yourself

1. Find the hypotenuse of a right triangle with legs $8$ and $15$.
2. Sides $9, 40, 41$ , right triangle? Justify.
3. A ladder $10$ m long rests against a wall such that its foot is $6$ m from the wall. How high does it reach?
4. In an isoceles right triangle, both legs are $a$. Find the hypotenuse.
5. Are $3, 5, 7$ the sides of a right triangle?
6. In $\triangle ABC$ with $\angle C = 90^\circ$, $a = 6, b = 8$. Find the perimeter and the area.
7. The lengths of two sides of a right triangle are $5$ and $13$. Find all possible lengths of the third side.
8. Show that the diagonal of a square of side $a$ is $a \sqrt{2}$.
9. The height of an equilateral triangle of side $a$ is $\dfrac{a \sqrt{3}}{2}$. Derive this using Pythagoras.
10. A triangle has sides $a, b, c$ with $a^2 + b^2 < c^2$. What can you say about the angle opposite $c$?

Pitfalls / Insight

• Identify the hypotenuse. It is always opposite the right angle, and it is always the longest side.
• For the converse, the square sum equals the square of the longest side, not just any side.
• Acute vs obtuse extension: if $a^2 + b^2 > c^2$, the angle opposite $c$ is acute; if $<$, obtuse.

Insight. Pythagoras is similarity in disguise: the altitude from the right angle splits the big right triangle into two smaller similar ones. Add up their squared legs, and the hypotenuse pops out.