M5 Find right-triangle sides and angles using ratios and Pythagoras

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Use Pythagoras and trigonometric ratios to solve right-triangle problems.

A right triangle has one angle of 90°. Its side lengths and angles are connected. You can use Pythagoras when you know two side lengths. You can use trigonometric ratios when you know an acute angle and a side, or when you know two sides and need an angle. This lesson reviews the needed triangle vocabulary, shows how to choose a method, and works through one example.

What you will learn

Identify the hypotenuse and the opposite and adjacent sides of a marked angle.
Use the Pythagorean theorem to find a missing side in a right triangle.
Choose a sine, cosine, or tangent ratio to find a missing side or angle.
Check that an answer makes sense and round it as requested.

1. Review: name the sides before calculating

An angle is measured in degrees. A right angle measures 90°. The other two angles in a right triangle are acute, which means each is less than 90°. Together, those two acute angles add to 90°.

The hypotenuse is the side opposite the 90° angle. It is always the longest side. To name the other two sides, first choose one of the acute angles. The opposite side is directly across from that angle. The adjacent side touches that angle and is not the hypotenuse.

Opposite and adjacent depend on which acute angle you choose. The hypotenuse does not change. Mark the right angle and the angle you are using before labelling sides. This prevents a common mix-up.

For example, picture a right triangle with a 90° corner at the bottom left and a chosen angle at the bottom right. The sloping side across from the 90° corner is the hypotenuse. The vertical side across from the chosen angle is opposite. The bottom side beside the chosen angle is adjacent.

The hypotenuse is opposite the right angle.
Opposite and adjacent are named in relation to a chosen acute angle.
Sketching and labelling the triangle is a useful first step.

2. Use Pythagoras when two sides are known

The Pythagorean theorem connects the three side lengths of a right triangle. It says that the square of the hypotenuse equals the sum of the squares of the other two sides. A square of a number means multiplying it by itself.

In the equation,

c

represents the hypotenuse, while

a

and

b

represent the two shorter sides. If the hypotenuse is missing, add the squares of the shorter sides and take the square root. If one shorter side is missing, subtract the known shorter side's square from the hypotenuse's square, then take the square root.

A square root reverses squaring. For instance, the positive square root of 25 is 5. A side length is positive, so use the positive root in a triangle problem.

Pythagoras applies only to right triangles. Before using it, check that one angle is 90°. It finds a missing side, not a missing angle. If the question asks for an angle, use a trigonometric ratio instead.

a^2+b^2=c^2

Use Pythagoras when two side lengths are known and a third side is needed.
The hypotenuse must be identified correctly.
A missing angle requires a ratio, not Pythagoras.

3. Use ratios to connect sides and angles

A ratio compares two quantities by division. In a right triangle, the sine, cosine, and tangent ratios compare side lengths with respect to a chosen acute angle. They are often shortened to sin, cos, and tan. These ratios work for angles measured in degrees.

Use the side labels from the chosen angle. Sine compares opposite with hypotenuse. Cosine compares adjacent with hypotenuse. Tangent compares opposite with adjacent. The memory aid SOH CAH TOA matches the first letters: Sine is Opposite over Hypotenuse; Cosine is Adjacent over Hypotenuse; Tangent is Opposite over Adjacent.

Choose the ratio that includes the side you know and the side you need. For example, if the opposite side and hypotenuse are involved, use sine. If the adjacent side and hypotenuse are involved, use cosine. If the opposite and adjacent sides are involved, use tangent. Then substitute the known angle and side lengths.

To find an angle when two side lengths are known, use the matching inverse ratio on a calculator. Inverse sine is written

ext{sin}^{-1}

, inverse cosine as

ext{cos}^{-1}

, and inverse tangent as

ext{tan}^{-1}

. These buttons return an angle; they do not mean one divided by sine, cosine, or tangent. Set the calculator to degree mode.

The table shows which sides belong in each ratio. Use it after choosing and marking the angle. Round only at the end when possible, because rounding early can slightly change the final result.

\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}

SOH CAH TOA helps select the correct ratio.
Use an inverse ratio to find an angle from two side lengths.
Check calculator degree mode and include units such as cm or degrees.

4. A solving plan and practice

Read the question and mark the right angle. Label the hypotenuse, then label opposite and adjacent using the angle in the question. Decide whether a side or an angle is missing. For a missing side with two other sides known, consider Pythagoras. For a missing side or angle connected to a known acute angle, choose the appropriate trigonometric ratio.

After calculating, ask whether the result is reasonable. The hypotenuse must be longer than either shorter side. Each acute angle must be between 0° and 90°, and the two acute angles must add to 90°. Include units and round to the precision requested.

Independent practice: A right triangle has legs of 8 cm and 15 cm. Find its hypotenuse. Then consider a different right triangle with an angle of 38° and a hypotenuse of 12 cm. Decide which ratio would find the side opposite the 38° angle. Finally, if a right triangle has opposite and adjacent sides of 7 and 9, decide which inverse ratio would find the angle.

Label first, choose a method second, and calculate third.
Estimate whether the answer fits the triangle before accepting it.
Keep full calculator values until the final rounding step.

Which sides go in each ratio?

Ratio	Side comparison	Use it when
Sine	Opposite ÷ hypotenuse	The opposite side and hypotenuse are involved
Cosine	Adjacent ÷ hypotenuse	The adjacent side and hypotenuse are involved
Tangent	Opposite ÷ adjacent	The opposite and adjacent sides are involved

Worked example

Find a side and then an angle

A right triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other shorter side. Then find the acute angle opposite the 5 cm side. Round the angle to the nearest degree.

Find the missing side
The 13 cm side is the hypotenuse, and the 5 cm side is one shorter side. Use Pythagoras to find the remaining shorter side, called $x$ . Subtract the known shorter side's square from the hypotenuse's square. Then take the positive square root because a side length is positive.
$x=\sqrt{13^2-5^2}=12\text{ cm}$
Choose a ratio for the angle
For the angle opposite the 5 cm side, the side of length 5 cm is opposite, and the side of length 12 cm is adjacent. Since these are the two sides being compared, use tangent.
$\tan(\theta)=\frac{5}{12}$
Calculate and round
Use inverse tangent on the calculator in degree mode. The calculator gives an angle close to 22.6°. Rounded to the nearest degree, the angle is 23°.
$\theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 23^\circ$

Answer: The missing side is 12 cm, and the angle opposite the 5 cm side is approximately 23°.

Check: The side lengths 5, 12, and 13 satisfy

5^2+12^2=13^2

. The other acute angle is about 67°, so the two acute angles add to 90°.

Common mistakes and how to avoid them

Calling the side beside the chosen angle the hypotenuse.

Correction: Find the side opposite the 90° angle first. That side is always the hypotenuse.

Using opposite and adjacent without stating which angle is being considered.

Correction: Mark the chosen acute angle before naming the sides. Opposite and adjacent depend on that angle.

Choosing a ratio based on a memorized word alone.

Correction: Identify the known and needed sides, then select the ratio that contains both.

Using a regular ratio instead of an inverse ratio to find an angle.

Correction: When side lengths are known and the angle is unknown, use the matching inverse ratio on the calculator.

Using Pythagoras in a triangle that is not a right triangle.

Correction: Confirm that the triangle has a 90° angle before applying the Pythagorean theorem.

Lesson summary

Name the hypotenuse from the right angle, then name opposite and adjacent from the chosen acute angle.
Use Pythagoras to find a missing side when the other two sides of a right triangle are known.
Use sine, cosine, or tangent for a side-angle relationship, and an inverse ratio to find an angle.
Check the result, use the correct units, and round as requested.

Check your understanding

Question 1

A right triangle has shorter sides of 9 cm and 12 cm. What is its hypotenuse?

15 cm
21 cm
3 cm
108 cm

Show answer and explanation

15 cm

Pythagoras gives the hypotenuse as the square root of

9^2+12^2

, which is 15 cm.

Question 2

Relative to angle

\theta

, which ratio compares the opposite side with the hypotenuse?

Cosine
Tangent
Sine
Pythagoras

Show answer and explanation

Sine

Sine compares the opposite side with the hypotenuse.

Question 3

A right triangle has opposite and adjacent sides of 6 cm and 8 cm relative to an unknown angle. Which calculation finds the angle?

$\tan^{-1}\!\left(\frac{6}{8}\right)$
$\sin^{-1}\!\left(\frac{6}{8}\right)$
$\cos^{-1}\!\left(\frac{6}{8}\right)$
$\tan\!\left(\frac{6}{8}\right)$

Show answer and explanation

\tan^{-1}\!\left(\frac{6}{8}\right)

Tangent compares opposite with adjacent. Since the angle is unknown, use inverse tangent.

Key terms

Hypotenuse: The side opposite the 90° angle in a right triangle.
Opposite side: The side directly across from a chosen acute angle.
Adjacent side: The side beside a chosen acute angle that is not the hypotenuse.
Ratio: A comparison of two quantities by division.
Inverse ratio: A calculator operation, such as inverse tangent, used to find an angle from side lengths.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M5. It is a study resource, not an official curriculum publication.

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M5 · Find right-triangle sides and angles using ratios and Pythagoras

Ontario Grade 10 Mathematics

Use Pythagoras and trigonometric ratios to solve right-triangle problems.

What you will learn

1. Review: name the sides before calculating

2. Use Pythagoras when two sides are known

3. Use ratios to connect sides and angles

4. A solving plan and practice

Which sides go in each ratio?

Find a side and then an angle

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MFM2P

About this lesson