1.01 Review: The Pythagorean theorem
The Pythagorean Theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written algebraically.
$a^2+b^2=c^2$a2+b2=c2
where $c$c represents the length of the hypotenuse and $a$a, $b$b are the two shorter sides. To see why this is true you can check out this investigation.
We can use the formula to find any side if we know the lengths of the two others.
Worked example
question 1
Find the length of the hypotenuse of a right triangle whose two other sides measure $3$3 cm and $4$4 cm.
Think: Here we want to find $c$c, and are given $a$a and $b$b.
Do: We will substitute the values we know in the formula and then solve to find $c$c.
| $c^2$c2 | $=$= | $3^2+4^2$32+42 | fill in the values for $a$a and $b$b |
| $c^2$c2 | $=$= | $9+16$9+16 | evaluate the squares |
| $c^2$c2 | $=$= | $25$25 | add the numbers together |
| $c$c | $=$= | $\sqrt{25}$√25 | take square root of both sides |
| $c$c | $=$= | $5$5 cm |
Reflect: Because all the lengths for the sides of the triangle are whole numbers, the numbers form a Pythagorean Triple, you can read more about the triples here.
Applying this relationship
If we need to find one of the shorter side lengths ($a$a or $b$b), using the formula we will have 1 extra step of rearranging to consider.
Worked example
question 2
Find the length of unknown side $b$b of a right triangle whose hypotenuse is $10$10 mm and one other side is $6$6 mm.
Think: Here we want to find $b$b, the length of a shorter side.
Do:
| $c^2$c2 | $=$= | $a^2+b^2$a2+b2 | start with the formula |
| $10^2$102 | $=$= | $6^2+b^2$62+b2 | fill in the values we know |
| $b^2$b2 | $=$= | $10^2-6^2$102−62 | rearrange to get the $b^2$b2 on its own |
| $b^2$b2 | $=$= | $100-36$100−36 | evaluate the right-hand side |
| $b^2$b2 | $=$= | $64$64 | |
| $b$b | $=$= | $8$8 | take the square root of both sides |
Reflect: Again, we can see that because all the lengths for the sides of the triangle are whole numbers, the numbers form a Pythagorean Triple.
Question 3
Find the length of unknown side $b$b of a right triangle whose hypotenuse is $6$6 mm and one other side is $4$4 mm.
Think: Here we want to find $b$b, the length of a shorter side.
Do:
| $c^2$c2 | $=$= | $a^2+b^2$a2+b2 | start with the formula |
| $6^2$62 | $=$= | $4^2+b^2$42+b2 | fill in the values we know |
| $b^2$b2 | $=$= | $6^2-4^2$62−42 | rearrange to get the $b^2$b2 on its own |
| $b^2$b2 | $=$= | $36-16$36−16 | evaluate the right-hand side |
| $b^2$b2 | $=$= | $20$20 | |
| $b$b | $=$= | $\sqrt{20}$√20 | take the square root of both sides |
| $b$b | $=$= | $4.47$4.47 mm | round to the required number of decimal places |
Reflect: This gives us a triangle with hypotenuse $6$6 mm and side lengths $4$4 and $4.47$4.47 mm. It's good to check at the end that you haven't ended up with a side length longer than the hypotenuse, as the hypotenuse has to be the longest length.
| $a^2+b^2$a2+b2 | $=$= | $c^2$c2 |
| other side lengths | hypotenuse |
The value $c$c is used to represent the hypotenuse which is the longest side of the triangle. The other two lengths are $a$a, $b$b.
Use the letters provided to you in the questions, if no letters are provided you can use $a$a and $b$b for either of the sides.
The converse of the Pythagorean theorem
We can also check to see if a triangle is indeed a right triangle by using the converse of the Pythagorean theorem. That is:
If $a^2+b^2=c^2$a2+b2=c2, then $\triangle ABC$△ABC is indeed a right triangle.
Practice questions
Question 4
Find the length of the hypotenuse, $c$c in this triangle.
Question 5
Calculate the value of $b$b in the triangle below. 
Question 6
Calculate the value of $a$a in the triangle below.
A right-angled triangle with the right angle at the bottom right corner, indicated by a small square symbol. The side opposite the right angle is the hypotenuse and measures 17 cm. The horizontal leg of the right angle, and also the base of the triangle, measures 15 cm. The vertical leg of the right angle, and also the height of the triangle, measures $a$a cm.