The Converse of Pythagoras’ Theorem

The theorem turned around

Pythagoras’ theorem starts with a right angle: if a triangle has a right angle, then the squares of the two shorter sides add up to the square of the longest side, $a^2+b^2=c^2$a² + b² = c².

The converse of a statement swaps its two parts. The converse of Pythagoras’ theorem starts with the three lengths: if $a^2+b^2=c^2$a² + b² = c², then the triangle has a right angle, and it is the angle across from the longest side c.

A converse is not true just because the statement is. "If a number is a multiple of 4, then it is even" is true, but its converse, "if a number is even, then it is a multiple of 4", is false: 6 is even and is not a multiple of 4. So the converse of Pythagoras’ theorem needs its own reason.

Two sides on a hinge

Join a side of 3 and a side of 4 at one end, with a hinge, so that the angle between them can change. The third side of the triangle joins their free ends.

Set the hinge at 90°. Now the triangle is a right triangle, so Pythagoras’ theorem gives the third side: $3^2+4^2=9+16=25=5^2$3² + 4² = 9 + 16 = 25 = 5², so it is 5.

The sides 3 and 4 meet at a right angle, and the third side is 5.

Close the hinge, then open it

Close the hinge to 70°. The free ends move closer together, and the third side shortens to about 4.1, which is less than 5.

Open the hinge to 110°. The free ends move farther apart, and the third side grows to about 5.8, which is more than 5.

The sides 3 and 4 on a hinge, with a square on every side. At 70° the square on the third side is less than 25, and the third side is about 4.1. Past 90° it is more than 25. Turn the hinge until the square on the third side is exactly 25.

Only 90° gives 5

As the hinge opens, the free ends move steadily apart, so the third side grows steadily. Nearly closed, the third side is just over 4 − 3 = 1. Nearly flat, it is just under 3 + 4 = 7. On the way it passes every length in between, each one exactly once, and it is 5 at 90°.

So a triangle with sides 3, 4 and 5 can only be the one with the hinge at 90°: the angle between the sides 3 and 4 is a right angle.

The same argument works for any two sides a and b. At 90° the third side is the length whose square is $a^2+b^2$a² + b², and at every other angle it is shorter or longer. So if the three sides of a triangle satisfy $a^2+b^2=c^2$a² + b² = c², the angle between a and b is a right angle. That is the converse of Pythagoras’ theorem.

The hinge also says which way a triangle misses. If $c^2$c² is less than $a^2+b^2$a² + b², the third side is shorter than at 90°, so the angle across from it is less than 90°. If $c^2$c² is more than $a^2+b^2$a² + b², the angle is more than 90°.

Testing three lengths

A triangle has sides 8, 15 and 17. The longest side is 17. Square the other two and add: $8^2+15^2=64+225=289$8² + 15² = 64 + 225 = 289. Square the longest side: $17^2=289$17² = 289. The two results are equal, so the triangle has a right angle, between the sides 8 and 15, across from the 17.

A triangle has sides 7, 8 and 11. Then $7^2+8^2=49+64=113$7² + 8² = 49 + 64 = 113, and $11^2=121$11² = 121. These are not equal, so there is no right angle. 121 is more than 113, so the angle across from the 11 is more than 90°.

A builder uses the converse to set out a square corner. From the corner, she marks 3 m along one wall and 4 m along the other. If the marks are exactly 5 m apart, the corner is a right angle.

The usual mistakes

Comparing the lengths instead of their squares. 8 + 15 = 23 is not 17, but that says nothing about the angle. The test is $8^2+15^2=289=17^2$8² + 15² = 289 = 17².

Squaring the wrong pair. The two shorter sides are squared and added, and the total is compared with the square of the longest side. $8^2+17^2=353$8² + 17² = 353 is not $15^2=225$15² = 225, but that is the wrong test: 17 is the longest side, so it is the one on its own.