Calculating the Hypotenuse
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 the lesson here.
We've already been looking at finding the length of the hypotenuse here, and in that set, all the answers ended up being whole number values. This meant that the sides of the triangles were forming Pythagorean Triples. That isn't always the case, so now we look at finding the hypotenuse in other cases, we will need to remember how to round to the required number of decimal places.
| $a^2+b^2$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.
When you have to round to a required number of decimal places, look to the digit in the following place value column. If that value is 5 or higher, we round up. If that value is less than 5 we round down. This lesson will help you with rounding decimals if you need a refresher.
Examples
Question 1
Find the length of the hypotenuse of a right-angled triangle whose two other sides measure $10$10 cm and $12$12 cm.
Think: Here we want to find $c$c, and are given $a$a and $b$b.
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 $2$2 decimal places |
Question 2
Calculate the value of $c$c in the triangle below. Round your answer to two decimal places.