Grade 8 The Theorem of Pythagoras

State and test the theorem

Three side lengths can tell you, without a protractor, whether a corner is truly square.

right angle

A square corner, like the corner of a page. It measures $90^{\circ }$.

right-angled triangle

A triangle with one right angle in it.

hypotenuse

The side opposite the right angle. It is always the longest side of the triangle.

square

A number multiplied by itself. The square of $5$ is $25$, written $5^2$.

Draw any right-angled triangle and measure its three sides. Now square each length. Every time, the two smaller squares add up to the biggest one. That is the Theorem of Pythagoras.

Do not believe it after one triangle. Try sides of $3$, $4$ and $5$. Then try $6$, $8$ and $10$. The rule holds both times, and it holds for every right-angled triangle.

The longest side has a name: the hypotenuse. Find the right angle first, then look straight across from it. The side you land on is the hypotenuse, however the triangle is turned. The other two are the shorter sides.

You can also work backwards. Given three lengths, square all three. If the two smaller squares add up to the biggest, the triangle is right-angled. If they do not, it is not. So three lengths are enough to test a corner.

Rules to remember

• $a^2+b^2=c^2$, with $c$ the hypotenuse

Examples

Traps

• Deciding the rule is true after trying it on one triangle. Test it on several right-angled triangles. One case is not proof.
• Picking a side next to the right angle as the hypotenuse. The hypotenuse is always the side directly opposite the right angle, and it is the longest side.
• Adding the two shorter sides and comparing that total with the longest side. Square each side first. $3+4=7$, but the test needs $3^2+4^2=25$.
• Testing the first side written down instead of the longest one. Sort the three lengths in your head. Only the longest can be the hypotenuse.

Learn and practise “State and test the theorem” in the app →

Find a missing side

If you know two sides of a right-angled triangle, the third one is already decided.

square root

The number that multiplies by itself to give you the one you started with. The square root of $25$ is $5$.

surd form

An answer left under the square root sign, because no exact decimal or fraction equals it.

Two sides are known and one is missing. Ask one question first: is the missing side the hypotenuse, or one of the shorter sides? Everything else follows from the answer.

If the hypotenuse is missing, add. Square both shorter sides, add the two answers, then take the square root. That last step is the one most often forgotten.

If a shorter side is missing, subtract. Square the hypotenuse, take away the square of the side you know, then take the square root. Take the smaller square from the bigger one, never the other way round.

Some square roots are not whole numbers. $\sqrt{29}$ sits between $5$ and $6$, and its decimals never stop. So leave it written as $\sqrt{29}$. That is surd form, and it is the exact answer.

In a word problem, hunt for the right angle first. A wall meets the ground at a right angle, so a ladder leaning on it is the hypotenuse.

Rules to remember

• $a^2+b^2=c^2$, with $c$ the hypotenuse

Examples

Traps

• Adding the two given sides instead of adding their squares. Square each side first, then add. $a^2+b^2=c^2$, not $a+b=c$.
• Stopping at the sum of the squares and calling that the hypotenuse. That total is the hypotenuse squared. Take its square root: $\sqrt{100}=10$.
• Adding the squares when the missing side is a shorter side, not the hypotenuse. If the hypotenuse is already known, subtract the other square from it instead of adding.
• Applying the theorem to a triangle that has no right angle. Check the triangle has a right angle marked before you use the theorem at all.

Learn and practise “Find a missing side” in the app →