Solve applied right-triangle problems

Ontario Grade 11 Mathematics

Geometry and Trigonometry

Model a practical situation, choose a useful relationship, and interpret the result.

A right triangle has one angle measuring

90^\circ

. It can model practical situations such as a ladder leaning against a wall or a person looking up at a tree. To solve an applied problem, sketch the situation, label the measurements you know, and identify what you need to find. Then choose a relationship that uses the information given. This lesson reviews the Pythagorean theorem and explains how sine, cosine, and tangent connect an angle to side lengths.

What you will learn

Represent a practical situation with a right triangle and label its sides.
Choose the Pythagorean theorem or a trigonometric ratio to find a missing measurement.
Use a calculator in degree mode and report a result with sensible units and precision.

1. Model the situation with a right triangle

Begin with a sketch. Mark the right angle and add any known measurements. For a ladder leaning against a vertical wall, the wall and the ground form the two sides meeting at the right angle. The ladder is the sloping side.

The hypotenuse is the side opposite the right angle. It is the longest side of a right triangle. Choose one of the other angles as your reference angle. The side across from it is the opposite side. The shorter side touching it is the adjacent side.

Opposite and adjacent depend on the reference angle. The hypotenuse does not. Label the sides after choosing the angle that matters to the question.

Keep units consistent. If measurements use different units, convert them before calculating. A good model should match the situation and show which length or angle the answer represents.

Sketch the situation and mark the right angle.
Choose a reference angle before labelling opposite and adjacent.
Keep measurement units consistent.

2. Use the Pythagorean theorem for side lengths

When you know two side lengths in a right triangle and need the third, use the Pythagorean theorem. It relates the two shorter sides to the hypotenuse. In the formula,

a

and

b

represent the shorter sides, and

c

represents the hypotenuse.

For example, if the two sides that meet at the right angle are

3

m and

4

m, the hypotenuse is

5

m. If the unknown is a shorter side, subtract the square of the known shorter side from the square of the hypotenuse. Then take the positive square root. A physical length is positive.

Check that you have identified the hypotenuse correctly before substituting. Keep enough digits during calculation and round the final measurement to a precision that suits the problem.

a^2+b^2=c^2

Use the theorem when the triangle is right-angled and two side lengths are known.
The hypotenuse is opposite the right angle.
Take the positive square root when finding a length.

3. Use a trigonometric ratio when an angle is known

A trigonometric ratio compares two side lengths in a right triangle. The three ratios used here are sine, cosine, and tangent. The memory aid SOH-CAH-TOA helps match each ratio to the sides: sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, and tangent uses opposite and adjacent.

Choose the ratio that includes both the side you know and the side you need. For example, if you know an acute angle and the adjacent side, and need the opposite side, tangent matches those sides. Substitute the known values and solve for the unknown using multiplication or division.

If the angle is unknown and two sides are known, use the matching inverse calculator function, such as inverse tangent. Set the calculator to degree mode when the angle is given in degrees. Inverse functions, often shown as sin⁻¹, cos⁻¹, and tan⁻¹, find an angle from a ratio; they do not mean taking the reciprocal. An acute angle in a right triangle is greater than

0^\circ

and less than

90^\circ

.

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

Sine compares opposite with hypotenuse; cosine compares adjacent with hypotenuse; tangent compares opposite with adjacent.
Choose a ratio that uses the known and unknown sides.
Use degree mode for angles measured in degrees.

4. Solve, interpret, and check

A clear solution includes a sketch, labelled sides, a chosen relationship, and a calculation. State the result with units and connect it to the original situation.

Use the Pythagorean theorem when two side lengths are known and you need the third. Use a trigonometric ratio when an acute angle and side information connect the measurement you know to the one you need. Choose the method that uses the given information directly.

Check whether the result fits the triangle and the situation. The hypotenuse must be longer than either shorter side. If a ramp rises above the ground, its sloping length must be longer than its horizontal run. A check can reveal a side-label mistake, a calculator setting error, or a calculation entry error.

Match the method to the information given and the measurement requested.
Include units and explain what the answer measures.
Check that the result is reasonable for the triangle and situation.

Worked example

A ladder reaches a wall

A

6.5

m ladder rests against a vertical wall. Its foot is

2.5

m from the wall. How high up the wall does it reach? Assume the wall and ground meet at a right angle.

Model the situation
The wall height and ground distance meet at the right angle. The ladder is opposite that angle, so it is the hypotenuse. The unknown height is a shorter side.
Choose a relationship
The triangle is right-angled, and the ladder and ground distance are known. Use the Pythagorean theorem, subtracting the square of the ground distance from the square of the hypotenuse to find the square of the height.
$h^2+2.5^2=6.5^2$
Calculate the height
Take the positive square root because a physical height is positive. The calculation gives $6.0$ m.
$h=\sqrt{6.5^2-2.5^2}=\sqrt{36}=6.0$

Answer: The ladder reaches

6.0

m up the wall.

Check: The ladder is the hypotenuse and is longer than either shorter side. The height is greater than the

2.5

m ground distance, which is reasonable for these measurements.

Worked example

Estimate a tree’s height above eye level

A student stands

18

m from the base of a tree on level ground. The angle of elevation from the student’s eye level to the top of the tree is

34^\circ

. How far above the student’s eye level is the top? Round to the nearest tenth of a metre.

Identify the sides
The ground distance is adjacent to the $34^\circ$ angle. The height above eye level is opposite that angle. The question does not ask for the sloping line of sight.
Select tangent
Tangent compares the opposite side with the adjacent side. These are the unknown height and the known ground distance, so tangent uses the information directly.
$\tan 34^\circ=\frac{h}{18}$
Solve and round
Multiply both sides by $18$ to isolate the height. In degree mode, the calculator gives about $12.1$ m to the nearest tenth.
$h=18\tan 34^\circ\approx 12.1$

Answer: The top of the tree is about

12.1

m above the student’s eye level.

Check: Since

34^\circ

is less than

45^\circ

, the opposite side is shorter than the adjacent side. The result,

12.1

m, is less than

18

m, which is reasonable.

Common mistakes and how to avoid them

Using the same opposite and adjacent labels for every angle.

Correction: Choose the reference angle first. Opposite and adjacent are named relative to that angle; the hypotenuse is always opposite the right angle.

Using the Pythagorean theorem on a triangle that is not right-angled.

Correction: Confirm that the model has a

90^\circ

angle before using the theorem.

Choosing a trigonometric ratio without labelling the sides.

Correction: Label the sides relative to the given angle, then choose the ratio containing the known and unknown sides.

Using the wrong calculator setting or rounding too early.

Correction: Use degree mode for angles given in degrees. Keep calculator precision until the final rounding step.

Lesson summary

Sketch and label the right triangle before calculating.
Use the Pythagorean theorem to connect the side lengths.
Use sine, cosine, or tangent when an angle and side lengths are connected.
Report the answer with units and check that it makes sense.

Check your understanding

Question 1

A right triangle has an acute angle of

28^\circ

. The side opposite that angle is

7

cm. Which equation can be used to find the hypotenuse,

c

?

$\sin 28^\circ=\frac{7}{c}$
$\cos 28^\circ=\frac{7}{c}$
$\tan 28^\circ=\frac{7}{c}$
$7^2+c^2=28^2$

Show answer and explanation

\sin 28^\circ=\frac{7}{c}

Sine compares the opposite side with the hypotenuse. Here, those sides are

7

cm and

c

, respectively.

Question 2

A right triangle has shorter sides of

5

m and

12

m. What is its hypotenuse?

$7$ m
$13$ m
$17$ m
$60$ m

Show answer and explanation

13

m

The Pythagorean theorem gives

c=\sqrt{5^2+12^2}=\sqrt{169}=13

m.

Key terms

Right triangle: A triangle with one angle measuring $90^\circ$ .
Hypotenuse: The side opposite the right angle; it is the longest side of a right triangle.
Opposite side: The side across from the chosen acute angle.
Adjacent side: The shorter side that touches the chosen acute angle.
Trigonometric ratio: A comparison of two side lengths in a right triangle, such as sine, cosine, or tangent.
Angle of elevation: The angle measured upward from a horizontal line to an object above it.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation C2.1. 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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