The side-length relationship for right triangles 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.
The side-length relationship for right triangles is sometimes called the Pythagorean theorem. However, we now know that Pythagoras was not actually the first person to use this property. It was used hundreds of years earlier by the Babylonians and possibly in other cultures around that time as well. We will generally call it the side-length relationship for right triangles, but you may see references to the Pythagorean theorem which is the same concept.
We can use the formula to find the length of the hypotenuse of any right-angled triangle, as long as we know the lengths of the other two sides.
| $a^2+b^2$a2+b2 | $=$= | $c^2$c2 |
| shorter 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 represented by $a$a and $b$b.
Worked example
Find the length of the hypotenuse of a right-angled triangle whose two other sides measure $10$10 cm and $12$12 cm. Round your answer to two decimal places.
Think: Here we want to find $c$c, and are given $a$a and $b$b. We can substitute the known values for $a$a and $b$b into the side-length relationship for right triangles $a^2+b^2=c^2$a2+b2=c2 and then solve for $c$c.
Do:
| $c^2$c2 | $=$= | $a^2+b^2$a2+b2 |
Start with the formula |
| $c^2$c2 | $=$= | $10^2+12^2$102+122 |
Fill in the values for $a$a and $b$b |
| $c^2$c2 | $=$= | $100+144$100+144 |
Evaluate the squares |
| $c^2$c2 | $=$= | $244$244 |
Add the $100$100 and $144$144 together |
| $c$c | $=$= | $\sqrt{244}$√244 |
Take the square root of both sides |
| $c$c | $=$= | $15.62$15.62 cm |
Rounded to two decimal places |
Reflect: We can see that the formula gives us a hypotenuse length of $15.62$15.62 cm. This is larger than both our shorter sides, as we should expect. If we got a number smaller than one (or both) of the short sides we know we have made a mistake in our calculation.
Exact values and rounded values
In some cases, where the sides of the right-angled triangle form a set of three integers that satisfy the side-length relationship for right triangles, the exact length of the hypotenuse is an integer value. But for most cases we will end up with an irrational number, that is, a radical.
In the worked example above we were asked to round our answer to two decimal places. In most cases we will want to keep our answer as an exact value.
If we are asked to give an exact answer, or to answer as a radical, we can stop our working out when we arrive at a line of working such as $c=\sqrt{11}$c=√11, as this can not be simplified without losing some accuracy when it is rounded.
Practice questions
Question 1
Consider the right-angled triangle. Which of the following equations do the sides of this triangle satisfy? $c^2=9^2+12^2$c2=92+122 $c=9^2+12^2$c=92+122 $12^2=9^2+c^2$122=92+c2 $9^2=c^2+12^2$92=c2+122 Solve the equation to find the length of the hypotenuse. Enter each line of work as an equation.
Question 2
Find the length of the unknown side $c$c in the triangle below. Write each step of work as an equation and give the answer as a radical.
Question 3
Find the length of the unknown side $c$c in the triangle below. Write each step of work as an equation and give the answer as a radical.
Set of three integers that satisfy the side-length relationship for right triangles
As mentioned above, we often have to round our answers as the answer is an irrational number like $\sqrt{2}$√2 or $\sqrt{11}$√11. However, there are special sets of three numbers that are all integers and could be the side lengths of a right triangle, so work in the side-length relationship for right triangles. We call these sets of numbers set of three integers that satisfy the side-length relationship for right triangles. These are sometimes referred to as Pythagorean Triples as they are three numbers that satisfy the Pythagorean theorem. For example, $3,4,5$3,4,5 is a set of three integers that satisfy the side-length relationship for right triangles because:
| $3^2+4^2$32+42 | $=$= | $9+16$9+16 |
Evaluate the squares of the two smaller numbers |
| $=$= | $25$25 |
Simplify the addition |
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| and |
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| $5^2$52 | $=$= | $25$25 |
Evaluate the square of the larger number |
We can think of this as a right triangle where the hypotenuse has length $5$5 and the shorter two sides have lengths $3$3 and $4$4.
If we are given the two smaller numbers, we can find the third by squaring the two smaller numbers and adding them together. We can draw a triangle and label with sides if that helps.
Practice questions
Question 4
The two smallest numbers in a set of three integers that satisfy the side-length relationship for right triangles are $5$5 and $12$12.
Which number completes the set?
$169$169
A$13$13
B$17$17
C$14$14
D