Lesson

We have been using the Pythagorean theorem 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.

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 Pythagorean formula $a^2+b^2=c^2$`a`2+`b`2=`c`2 and then solve for $a$`a`.

**Do:** We will substitute $b=11$`b`=11 and $c=20$`c`=20 into the formula for the Pythagorean theorem:

$a^2+b^2$a2+b2 |
$=$= | $c^2$c2 |
Start with the formula |

$a^2+11^2$a2+112 |
$=$= | $20^2$202 |
Fill in the values for $b$ |

$a^2$a2 |
$=$= | $20^2-11^2$202−112 |
Subtract $11^2$112 from both sides to make $a^2$ |

$a^2$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.

Careful!

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).

We can also rearrange the equation before we perform the substitution, to find formulas for each side length.

Rearranging the Pythagorean theorem

To find a shorter side:

$a^2=c^2-b^2$`a`2=`c`2−`b`2 or $b^2=c^2-a^2$`b`2=`c`2−`a`2

We can take the square root of both sides to give us the following formulas:

$a=\sqrt{c^2-b^2}$`a`=√`c`2−`b`2 or $b=\sqrt{c^2-a^2}$`b`=√`c`2−`a`2

Consider the right-angled triangle.

Which of the following equations do the sides of this triangle satisfy?

$13^2=k^2-5^2$132=

`k`2−52A$k=13^2-5^2$

`k`=132−52B$k^2=13^2+5^2$

`k`2=132+52C$k^2=13^2-5^2$

`k`2=132−52DSolve the equation to find the length of the unknown side.

Enter each line of working as an equation.

Find the length of the unknown side $s$`s` in the triangle below.

Write each step of working as an equation and give the answer as a radical.

Find the length of the unknown side $k$`k` in the triangle below.

Write each step of working as an equation and give the answer to two decimal places.

Describe the Pythagorean relationship using various geometric models, and apply the theorem to solve problems involving an unknown side length for a given right triangle.