Pythagorean Theorem: Find Missing Sides, Fix Errors | Grade 8 Math | LevelKite

Every right triangle has one right angle, which is an angle of $90^\circ$ . You already know the names of its sides: the two legs form the right angle, and the hypotenuse is the side opposite the right angle.

In this lesson you will use one fact about right triangles to do three things:

find the hypotenuse when you know both legs,
find a leg when you know the hypotenuse and the other leg,
find and fix mistakes in a worked solution.

That one fact is about the areas of squares.

A square with side length $s$ has area $s \times s$ . You write this as $s^2$ and read it as " $s$ squared."

Each side of a right triangle can also be one side of a square. So a right triangle comes with three squares: one on each leg and one on the hypotenuse.

Take the right triangle with legs $3$ cm and $4$ cm. Its hypotenuse is exactly $5$ cm long. The table shows the area of the square on each side, in square centimeters.

Squares on the sides of the right triangle with legs 3 cm and 4 cm

Side	Length (cm)	Area of the square on that side (square cm)
Leg	$3$	$3^2 = 9$
Other leg	$4$	$4^2 = 16$
Hypotenuse	$5$	$5^2 = 25$

Look at the last column. The two leg squares together have an area of $9 + 16 = 25$ square centimeters. That is exactly the area of the square on the hypotenuse.

This is not special to this one triangle. It happens in every right triangle, and that fact is called the Pythagorean theorem. A theorem is a statement that has been proved true.

Key concept

The Pythagorean theorem

In a right triangle, let $a$ and $b$ be the lengths of the two legs and let $c$ be the length of the hypotenuse, the side opposite the right angle. Measure all three in the same unit, so $a$ , $b$ and $c$ are positive numbers.

Then the areas of the squares on the two legs add up to the area of the square on the hypotenuse: $a^2 + b^2 = c^2$ .

Two conditions always come with this rule:

The triangle must be a right triangle. You must know this from the problem, either in words or from the small square symbol that marks a right angle. A corner that only looks like a right angle is not enough.
$c$ must be the hypotenuse. A leg can never take the place of $c$ .

a^2 + b^2 = c^2

Finding the hypotenuse. In a right triangle, when you know both legs you know $a$ and $b$ . Adding their squares gives $c^2$ , the area of the square on the hypotenuse. But you want the side $c$ , not the area.

To go from a square's area back to its side, you use a square root. The square root of a number $n \ge 0$ , written $\sqrt{n}$ , is the nonnegative number (zero or positive) whose square is $n$ . For example, $\sqrt{25} = 5$ because $5^2 = 25$ and $5$ is not negative.

For the triangle in the table, $c^2 = 9 + 16 = 25$ . Two numbers have a square of $25$ : $5$ and $-5$ , because $(-5)^2 = 25$ as well. A side length cannot be negative, so you keep only the nonnegative root: $c = \sqrt{25} = 5$ , which matches the $5$ cm in the table.

Worked example

A hypotenuse with a whole-number length

A right triangle has legs of length $6$ cm and $8$ cm. Find the length of its hypotenuse.

The triangle is a right triangle, so the Pythagorean theorem applies. The legs are 6 cm and 8 cm, so a = 6 and b = 8. Let c stand for the length of the hypotenuse in centimeters.
$a^2 + b^2 = c^2$
Replace a with 6 and b with 8. The equation stays true because 6 and 8 are the leg lengths of this right triangle.
$6^2 + 8^2 = c^2$
Square each leg, which means multiply it by itself: 6 times 6 is 36, and 8 times 8 is 64. Writing a number in a simpler form does not change its value, so the equation is still true.
$36 + 64 = c^2$
Add 36 and 64 to get 100. The two sides of an equation can be written in either order, so put c squared first. The 100 is the area, in square centimeters, of the square on the hypotenuse.
$c^2 = 100$
The side c is a length, so it is the nonnegative square root of 100. That root is 10, because 10 times 10 is 100.
$c = \sqrt{100} = 10$

Answer

The hypotenuse is $10$ cm long.

Worked example

A hypotenuse that is not a whole number

A right triangle has legs of length $4$ m and $7$ m. Find the length of its hypotenuse. Give the exact length, then round it to the nearest hundredth of a meter.

The triangle is a right triangle and its legs are 4 m and 7 m, so a = 4 and b = 7. Let c stand for the length of the hypotenuse in meters. Put the leg lengths into the theorem.
$4^2 + 7^2 = c^2$
Square each leg and add: 4 times 4 is 16, 7 times 7 is 49, and 16 plus 49 is 65. Write c squared first, as before.
$c^2 = 16 + 49 = 65$
Take the nonnegative square root, because c is a length. No whole number has a square of 65, since 8 times 8 is 64 and 9 times 9 is 81. So leave the root as it is. This is the exact length.
$c = \sqrt{65}$
Use a calculator to write the root as a decimal. The digits keep going, so you round. The thousandths digit is 2, so round down to 8.06. A rounded value is not exact, so the sign changes from = to ≈, which means 'is approximately equal to'.
$\sqrt{65} = 8.0622\ldots \approx 8.06$
Check. Squaring the rounded value gives a number close to 65 but not equal to it, which shows that 8.06 is an approximation. The size check passes too: 8.06 is greater than both legs, 4 and 7.
$8.06^2 = 64.9636 \approx 65$

Answer

The hypotenuse is exactly $\sqrt{65}$ m long, which is about $8.06$ m.

Finding a leg. Sometimes you know the hypotenuse and one leg of a right triangle, and the other leg is missing. The theorem is the same, and so is the equation $a^2 + b^2 = c^2$ . What changes is where the unknown sits: it is now one of the two squares being added.

Think about the three squares again. The hypotenuse square has the same area as both leg squares together. So if you take one leg square away from the hypotenuse square, what is left is the other leg square.

In symbols, you subtract the known leg's square from both sides of the equation. Subtracting the same number from both sides keeps the two sides equal. Then you take the nonnegative square root, just as before.

Worked example

Finding a missing leg

A right triangle has a hypotenuse of length $13$ cm and one leg of length $5$ cm. Find the length of the other leg.

Sort the sides before you write the equation. The triangle is a right triangle and its hypotenuse is 13 cm, so c = 13 and its square stands alone on one side. The known leg is 5 cm, so a = 5. Let b stand for the length of the missing leg in centimeters.
$5^2 + b^2 = 13^2$
Square the two known lengths: 5 times 5 is 25, and 13 times 13 is 169. Each side keeps its value, so the equation is still true.
$25 + b^2 = 169$
Subtract 25 from both sides so that the missing square is alone. Taking the same number from both sides keeps them equal.
$b^2 = 169 - 25 = 144$
Take the nonnegative square root, because b is a length. The root is 12, since 12 times 12 is 144.
$b = \sqrt{144} = 12$
Check by putting 12 back into the first equation: both sides equal 169. The size check passes as well, because the leg, 12, is shorter than the hypotenuse, 13.
$5^2 + 12^2 = 25 + 144 = 169 = 13^2$

Answer

The other leg is $12$ cm long.

Finding and correcting errors. Everyone makes slips, so checking is part of solving. To check a solution, yours or someone else's, do not look only at the final answer. Read the work one line at a time.

Check the first line against the theorem and its two conditions. Then ask whether each later line follows from the line before it. The first line that fails is the error. To correct it, fix that line and redo every line after it, because the later lines were built on the mistake.

Common errors come in a few kinds. The table gives a question that catches each kind.

Six questions for checking work with the Pythagorean theorem

Question to ask	A mistake it catches	Why that is a mistake
Is the triangle a right triangle?	Using $a^2 + b^2 = c^2$ when no right angle is given	The theorem is only about right triangles.
Is the hypotenuse alone as $c$ ?	Putting a known hypotenuse in a leg's place and adding	$c$ must be the hypotenuse, the side opposite the right angle.
Is each length squared?	Doubling, such as writing $7^2 = 14$	$7^2$ means $7 \times 7 = 49$ , not $7 \times 2$ .
Is a square root taken at the end?	Finding $c^2 = 100$ and answering $c = 100$	$100$ is the area of the square. The side is $\sqrt{100} = 10$ .
Is the root taken of the whole total?	Writing $\sqrt{36 + 64} = 6 + 8 = 14$	Add first: $\sqrt{36 + 64} = \sqrt{100} = 10$ , and $10$ is not $14$ .
Is the size sensible?	A leg that comes out longer than the hypotenuse	In a right triangle the hypotenuse is longer than each leg.

Worked example

Finding and correcting an error

A right triangle has a hypotenuse of length $12$ m and one leg of length $8$ m. Dana finds the length of the other leg, $x$ meters, to the nearest hundredth, in four lines. Line 1: $8^2 + x^2 = 12^2$ . Line 2: $16 + x^2 = 24$ . Line 3: $x^2 = 8$ . Line 4: $x = \sqrt{8} \approx 2.83$ . Find Dana's error and correct it.

Check Line 1, the set-up. The triangle is a right triangle. The hypotenuse is 12 m, and its square stands alone on one side. The squares of the two legs are added on the other side. Line 1 is correct.
$8^2 + x^2 = 12^2$
Check Line 2, the squares. Squaring 8 means 8 times 8, which is 64. Dana wrote 16, which is 8 times 2. Likewise, 12 squared is 144, not 24. Dana doubled instead of squaring. This is the first line that fails, so it is the error.
$8^2 = 8 \times 8 = 64, \qquad 12^2 = 12 \times 12 = 144$
Correct Line 2 by using the true squares. Lines 3 and 4 were built on the wrong numbers, so redo them too.
$64 + x^2 = 144$
Subtract 64 from both sides, which keeps the two sides equal.
$x^2 = 144 - 64 = 80$
Take the nonnegative square root, because x is a length. The number 80 lies between 64, which is 8 times 8, and 81, which is 9 times 9, so the root is not a whole number. Keep the root as the exact value. For a decimal, round the calculator value: the thousandths digit is 4, so round down to 8.94.
$x = \sqrt{80} = 8.9442\ldots \approx 8.94$
Check the corrected value in Line 1. The left side is very close to 144, as it should be for a rounded value. Dana's answer, 2.83, was also shorter than the hypotenuse, so the size check alone would have missed this error. Reading line by line found it.
$8^2 + 8.94^2 = 64 + 79.9236 = 143.9236 \approx 144$

Answer

Dana doubled $8$ and $12$ instead of squaring them. The other leg is exactly $\sqrt{80}$ m long, which is about $8.94$ m.

Takeaway. In a right triangle with legs $a$ and $b$ and hypotenuse $c$ , the side opposite the right angle, the areas of the squares on the legs add up to the area of the square on the hypotenuse: $a^2 + b^2 = c^2$ .

Add the squares of the legs to get the hypotenuse's square. Subtract one leg's square from the hypotenuse's square to get the other leg's square. Either way, finish with the nonnegative square root, because a length cannot be negative. To find an error, check the two conditions first, then read the work line by line.

Next, you will use these steps to find perimeters and areas and to solve word problems.