Solve right-triangle side-length problems

Ontario Grade 9 Mathematics

Geometry and Measurement

Use the Pythagorean theorem to find a missing side

A rectangular garden has a path running diagonally from one corner to the opposite corner. The path and two edges of the garden form a right triangle. If you know the lengths of the two edges, you can find the path length. If you know the path and one edge, you can find the other edge. The Pythagorean theorem gives a way to solve both kinds of problems. A right triangle is a triangle with one square corner, called a right angle.

What you will learn

Identify the hypotenuse and the two shorter sides of a right triangle.
Use the Pythagorean theorem to find a missing side length.
Check that a calculated side length makes sense in the problem.

Recognize the sides of a right triangle

A side length is the distance from one corner of a shape to another. In a right triangle, one side is across from the right angle. This side is called the hypotenuse. The hypotenuse is always the longest side. The other two sides meet to form the right angle.

Before using the rule, recall two arithmetic ideas. Squaring a number means multiplying it by itself. For example,

5^2=5\times5=25

. A square root reverses squaring: since

5^2=25

, the square root of

25

is

5

.

A simple sketch can help you label the sides. Mark the right angle first. The side across from it is the hypotenuse. The other two sides are the legs. The letters used for side lengths can vary, so rely on their positions in the triangle, not only on a letter.

The hypotenuse is opposite the right angle and is the longest side.
The two legs meet at the right angle.
A squared length is the length multiplied by itself.

The side-length rule

The Pythagorean theorem links the three side lengths of every right triangle. The squares of the two leg lengths add to the square of the hypotenuse length. If the legs have lengths

a

and

b

, and the hypotenuse has length

c

, the rule is shown below.

Use the rule only after confirming that the triangle is right-angled. Label the hypotenuse as

c

in the equation, even if the problem uses a different letter for that side. This keeps the longest side in the correct place.

If both legs are known, add their squared lengths, then find the square root of the total to get the hypotenuse. If the hypotenuse and one leg are known, subtract the known leg's square from the hypotenuse's square. Then find the square root to get the other leg. Subtraction is needed because the known leg's square is one part of the total.

a^2+b^2=c^2

Add the leg squares to find the hypotenuse square.
Subtract a leg square from the hypotenuse square to find the other leg square.
Take the square root at the end to return to a side length.

Choose the correct calculation

Start by identifying which side is missing. If the missing side is opposite the right angle, it is the hypotenuse. If it meets the right angle, it is a leg.

For a missing hypotenuse, the equation has the form

a^2+b^2=c^2

. Add the two known squares and take the square root. For a missing leg, rearrange the relationship by subtracting the known leg square from the hypotenuse square. The hypotenuse must be longer than either leg, so a result that contradicts this is a sign to check the labels or arithmetic.

Lengths in a real-world problem are usually positive. Use the positive square root as the side length. If a calculator gives a decimal, round only as the problem asks. If no rounding direction is given, keep enough digits to show a useful answer and label the units.

a^2=c^2-b^2

Decide whether the unknown is a leg or the hypotenuse before calculating.
A leg is shorter than the hypotenuse.
Include the correct unit with a real-world length.

Check the result

A calculation is not finished until you ask whether the answer fits the triangle and the situation. For example, a diagonal across a rectangle must be longer than either edge. A missing leg must be shorter than the hypotenuse.

You can also substitute the side lengths into the Pythagorean theorem. Square each leg and add. The result should equal the hypotenuse squared, allowing for small differences if a decimal was rounded.

Keep the labels attached to the right sides throughout your work. Do not put the hypotenuse in the position of a leg. That changes the relationship and can lead to an incorrect result.

Check the answer against the side positions and the context.
Substitution can confirm that the side lengths satisfy the rule.
Check units and rounding as well as arithmetic.

Which calculation fits?

Missing side	Relationship to use	What to do
Hypotenuse $c$	$a^2+b^2=c^2$	Add the leg squares, then take the square root.
Leg $a$	$a^2=c^2-b^2$	Subtract the known leg square from the hypotenuse square, then take the square root.

Worked example

Find a missing leg

A rectangular sign is

13

centimetres tall and has a diagonal brace that is

15

centimetres long. What is the width of the sign?

Identify the sides
The height and width meet at a right angle, so they are the legs. The diagonal brace is across from that angle, so it is the hypotenuse. Let the unknown width be $w$ .
Set up the relationship
Put the two legs in the leg positions and the diagonal in the hypotenuse position. This gives an equation for the unknown width.
$w^2+13^2=15^2$
Square the known lengths
Squaring $13$ and $15$ changes the equation into one that can be solved by subtraction. The diagonal's square is the total of the two leg squares.
$w^2+169=225$
Find the unknown square
Subtract $169$ from both sides to leave the unknown squared by itself. Subtracting is appropriate because the known leg is one part of the total.
$w^2=56$
Find the width
Take the positive square root because a width is a positive length. The square root of $56$ is about $7.48$ , so the width is about $7.5$ centimetres to the nearest tenth.
$w=\sqrt{56}\approx7.48$

Answer: The sign is about

7.5

centimetres wide.

Check: The hypotenuse is

15

centimetres, longer than either leg. Also,

7.48^2+13^2

is approximately

225

, which matches

15^2

.

Common mistakes and how to avoid them

Putting the hypotenuse in the position of a leg.

Correction: Find the right angle first. The side across from it is the hypotenuse and belongs alone on the right side of

a^2+b^2=c^2

.

Adding squares when finding a missing leg.

Correction: When the hypotenuse and one leg are known, subtract the known leg's square from the hypotenuse's square.

Stopping at the squared side length.

Correction: After finding the unknown squared, take its positive square root to get the side length.

Rounding too early or leaving off units.

Correction: Keep the value unrounded until the final step, follow the requested rounding, and state the unit.

Lesson summary

A right triangle has one right angle, two legs, and a hypotenuse opposite the right angle.
The Pythagorean theorem is $a^2+b^2=c^2$ , with $c$ as the hypotenuse.
For a missing hypotenuse, add the squared legs. For a missing leg, subtract the known leg's square from the hypotenuse's square.
Take the positive square root and check that the result fits the triangle and the context.

Check your understanding

Question 1

A right triangle has legs of

6

centimetres and

8

centimetres. What is its hypotenuse?

$10$ centimetres
$14$ centimetres
$2$ centimetres
$100$ centimetres

Show answer and explanation

10

centimetres

The hypotenuse squared is

6^2+8^2=36+64=100

. Its positive square root is

10

.

Question 2

A right triangle has a hypotenuse of

17

metres and one leg of

8

metres. What is the other leg?

$15$ metres
$25$ metres
$9$ metres
$225$ metres

Show answer and explanation

15

metres

The missing leg squared is

17^2-8^2=289-64=225

. The positive square root of

225

is

15

.

Question 3

A right triangle has a hypotenuse of

9

centimetres and one leg of

12

centimetres. Which statement is correct?

These lengths cannot be sides of the same right triangle.
The missing leg is $3$ centimetres.
The missing leg is $15$ centimetres.
The missing leg is $\sqrt{225}$ centimetres.

Show answer and explanation

These lengths cannot be sides of the same right triangle.

The hypotenuse must be the longest side. A side of

12

centimetres cannot be a leg when the hypotenuse is only

9

centimetres.

Key terms

Right triangle: A triangle with one right angle.
Right angle: An angle that forms a square corner.
Leg: Either of the two sides that meet at the right angle.
Hypotenuse: The side opposite the right angle; it is the longest side.
Square root: A number that, when multiplied by itself, gives the number under the root sign.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation E1.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E1.5 · Solve right-triangle side-length problems

Ontario Grade 9 Mathematics

Use the Pythagorean theorem to find a missing side

What you will learn

Recognize the sides of a right triangle

The side-length rule

Choose the correct calculation

Check the result

Which calculation fits?

Find a missing leg

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MTH1W

About this lesson and its review