We have been using the side-length relationship for right triangles to relate the three sides of a right-angled triangle together:
Up until now we have been using this formula to find the length of the hypotenuse, knowing the length of the two short sides.
We can use the same formula to find the length of a short side, knowing the length of the hypotenuse and the length of the other short side. The only difference when finding a short side is that we can put the numbers in the wrong way around if we aren't careful.
Worked example
A right-angled triangle has a hypotenuse of length $20$20 m and one short side that has a length of $11$11 m. Find the exact length of the other short side, and find its length rounded to two decimal places.
Think: Since we want to find the length of a short side, we will be solving for either $a$a or $b$b- let's choose $a$a. Since we want to find $a$a, our given values will be $b$b and $c$c which we can substitute into the side-length relationship for right triangles $a^2+b^2=c^2$a2+b2=c2 and then solve for $a$a.
Do: We will substitute $b=11$b=11 and $c=20$c=20 into the formula for the side-length relationship for right triangles:
| $a^2+b^2$a2+b2 | $=$= | $c^2$c2 |
Start with the formula |
| $a^2+11^2$a2+112 | $=$= | $20^2$202 |
Fill in the values for $b$b and $c$c |
| $a^2$a2 | $=$= | $20^2-11^2$202−112 |
Subtract $11^2$112 from both sides to make $a^2$a2 the subject |
| $a^2$a2 | $=$= | $400-121$400−121 |
Evaluate the squares |
| $a^2$a2 | $=$= | $279$279 |
Subtract $121$121 from $400$400 |
| $a$a | $=$= | $\sqrt{279}$√279 m |
Take the square root of both sides |
| $a$a | $=$= | $16.70$16.70 m |
Rounded to two decimal places |
The exact length is $\sqrt{279}$√279 m, and the rounded length is $16.70$16.70 m.
Reflect: When finding a short side, our answer should always be shorter than the hypotenuse. If our answer is longer, we know we have made a mistake.
The most important thing to remember when finding a short side is that the two lengths need to go into different parts of the formula.
If you get the lengths around the wrong way, you will probably end up with the square root of a negative number (and a calculator error).
Rearranging to find a short side
We can also rearrange the equation before we perform the substitution, to find formulas for each side length.
To find a shorter side:
$a^2=c^2-b^2$a2=c2−b2 or $b^2=c^2-a^2$b2=c2−a2
We can take the square root of both sides to give us the following formulas:
$a=\sqrt{c^2-b^2}$a=√c2−b2 or $b=\sqrt{c^2-a^2}$b=√c2−a2
Practice questions
Question 1
Consider the right-angled triangle. Which of the following equations do the sides of this triangle satisfy? $k^2=13^2+5^2$k2=132+52 $k=13^2-5^2$k=132−52 $13^2=k^2-5^2$132=k2−52 $k^2=13^2-5^2$k2=132−52 Solve the equation to find the length of the unknown side. Enter each line of working as an equation.
Question 2
Find the length of the unknown side $s$s in the triangle below. Write each step of work as an equation and give the answer as a square root.
Question 3
Find the length of the unknown side $m$m in the triangle below. Write each step of work as an equation and give the answer to two decimal places.
Set of three integers that satisfy the side-length relationship for right triangles
Previously we have looked at the set of three integers that satisfy the side-length relationship for right triangles or Pythagorean Triples. In other words, we can think of the three numbers as the side lengths of a right triangle where the hypotenuse is the biggest number and the other two numbers are the shorter sides.
If we are given the largest number, $c$c, and one of the smaller numbers, $a$a, we can find the third subtracting the square of the smaller from the square of the bigger, $c^2-a^2$c2−a2. We can draw a triangle and label with sides if that helps.
You might find that you start to notice the first few as they are very common in questions.
- $3,4,5$3,4,5 and its multiples like $6,8,10$6,8,10 or $9,12,15$9,12,15
- $5,12,13$5,12,13 and its multiples like $10,24,26$10,24,26
- $8,15,17$8,15,17 and its multiples
- $7,24,25$7,24,25 and its multiples
Practice questions
Question 4
The two largest numbers in a set of three integers that satisfy the side-length relationship for right triangles are $15$15 and $12$12.
Which number completes the set?
$9$9
A$3$3
B$8$8
C$81$81
D