Solve practical problems with similar triangles

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Use matching triangle shapes and side ratios to find an unknown length

A triangle can help you measure something that is too tall or difficult to reach. For example, you can compare the shadow of a tree with the shadow of a person. If sunlight makes the shadows at the same angle, the two right triangles have the same shape. Their side lengths can then be compared. This lesson reviews ratios and proportions, explains how to identify matching sides, and uses them to find an unknown measurement.

What you will learn

Recognize when a real situation can be represented by similar triangles.
Match corresponding sides before setting up a proportion.
Solve a practical problem using a scale relationship and check whether the answer makes sense.

1. Grade 9 bridge: ratios and proportions

A ratio compares two quantities by division. For example, a height of

2

metres compared with a shadow of

3

metres can be written as

\frac{2}{3}

. Keep the order clear: height is compared with shadow in this ratio.

A proportion says that two ratios are equal. If two objects make triangles with the same shape, the ratio of a triangle’s height to its base will match the ratio for the other triangle. This lets you use measurements you know to find a length you do not know.

Similar triangles are triangles with the same shape. They may be different sizes. Their matching angles have equal measures, and their matching side lengths have the same scale relationship. A scale relationship means that all corresponding sides are multiplied by the same number.

A ratio compares quantities in a stated order.
Similar triangles have the same shape, but not necessarily the same size.
Corresponding sides are sides that occupy matching positions in the two triangles.

2. Finding similar triangles in a practical situation

Start with a sketch and label the measurements you know. In a shadow problem, the object’s height and its shadow form the two shorter sides of a right triangle. The slanted side represents the sun ray. The ground and the upright object meet at a right angle.

A right angle measures

90^\circ

. When sunlight reaches two upright objects, the sun rays are treated as parallel in the sketch. The rays make the same angle with the ground. Both triangles also have a right angle, so the triangles have the same shape.

Before writing a proportion, match the sides by their roles. Compare height with height, and shadow length with shadow length. Do not compare a height from one triangle with a slanted side from the other. A quick labelled sketch or a table can help keep the pairs straight.

In a shadow sketch, the upright height matches the upright height.
The ground shadow matches the ground shadow.
Similar shape is the reason the matching side ratios are equal.

3. Set up and solve a proportion

Let

h_1

and

h_2

be the heights of two objects, and let

s_1

and

s_2

be their shadow lengths. If the triangles are similar, the height-to-shadow ratio is the same for each one. Choose the order of the sides once, then keep that order in both ratios.

When one height is unknown, substitute the known measurements into the proportion. Then solve for the missing height using the same basic equation skills used with other proportions. If a variable is multiplied by a number, divide both sides by that number to leave the variable by itself.

Check that the answer has the right unit and is reasonable. For example, if the unknown object is taller than the measured person, its shadow should be longer when both objects are lit by the same sun. A result that conflicts with the situation may mean the sides were mismatched or the arithmetic needs checking.

\frac{h_1}{s_1}=\frac{h_2}{s_2}

Write ratios in the same order on both sides of the proportion.
Use consistent units for corresponding measurements.
Check the answer against the practical situation.

4. Guided example and independent practice

A student who is

1.6

m tall casts a shadow that is

2.1

m long. At the same time, a tree casts a shadow that is

8.4

m long. The person and tree make similar triangles because each object is upright, each shadow lies along level ground, and the sunlight has the same direction. The next example uses these measurements to find the tree’s height.

For independent practice, imagine a signpost casts a

3

m shadow while a nearby

1.5

m marker casts a

2

m shadow. Sketch the two right triangles, label which sides correspond, and write a proportion to find the signpost’s height. Check whether your result is greater or less than the marker’s height.

A sketch helps identify corresponding sides before calculation.
Independent practice: use height-to-shadow ratios for both triangles.

Match the sides in the shadow triangles

Side in student triangle	Corresponding side in tree triangle
Student height: $1.6$ m	Tree height: $h$ m
Student shadow: $2.1$ m	Tree shadow: $8.4$ m

Worked example

Estimate a tree’s height from its shadow

A student who is

1.6

m tall casts a shadow that is

2.1

m long. At the same time, a tree casts a shadow that is

8.4

m long. Estimate the tree’s height.

Identify matching sides
The student and tree are upright, and both shadows lie on the ground. The sunlight has the same direction for both triangles. So the student’s height corresponds to the tree’s height, and the student’s shadow corresponds to the tree’s shadow.
Set up the proportion
Compare height with shadow for each triangle. Let $h$ represent the tree’s height in metres. Keeping height first and shadow second gives equal ratios.
$\frac{1.6}{2.1}=\frac{h}{8.4}$
Solve for the height
Multiply both sides by $8.4$ to isolate $h$ . The calculation gives the tree’s height in metres.
$h=8.4(\frac{1.6}{2.1})=6.4$
Check the result
The tree’s shadow is four times as long as the student’s shadow. The calculated tree height is also four times the student’s height, so the matching side lengths have a consistent scale relationship.
$\frac{8.4}{2.1}=4, \frac{6.4}{1.6}=4$

Answer: The tree is about

6.4

m tall.

Check: The tree’s shadow is four times the student’s shadow, and the tree’s height is four times the student’s height.

Common mistakes and how to avoid them

Comparing the student’s height with the tree’s shadow.

Correction: Match sides by their roles. Compare height with height and shadow with shadow.

Reversing the order in only one ratio.

Correction: If one ratio is height divided by shadow, write the other ratio in the same order.

Using measurements with different units without changing them.

Correction: Convert measurements to the same unit before forming the proportion.

Assuming any two triangles in a real situation are similar.

Correction: Check that the situation gives the same triangle shape, such as upright objects, level ground, and sunlight travelling in the same direction.

Lesson summary

Similar triangles have the same shape, so corresponding sides share the same scale relationship.
In a shadow problem, match each height with the other height and each shadow with the other shadow.
Write equal ratios in the same order, solve for the unknown, and check that the result fits the situation.

Check your understanding

Question 1

A

1.2

m post casts a

1.5

m shadow. At the same time, a nearby object casts a

6

m shadow. Which proportion can be used to find the object’s height

h

?

$\frac{1.2}{1.5}=\frac{h}{6}$
$\frac{1.2}{1.5}=\frac{6}{h}$
$\frac{1.2}{6}=\frac{1.5}{h}$
$\frac{1.2}{h}=\frac{1.5}{6}$

Show answer and explanation

\frac{1.2}{1.5}=\frac{h}{6}

The post’s height corresponds to the unknown object’s height, and the post’s shadow corresponds to the longer shadow. Comparing height with shadow in both triangles gives the first proportion.

Question 2

Using the measurements in the question above, what is the object’s height?

$4.8$ m
$7.5$ m
$5.0$ m
$0.3$ m

Show answer and explanation

4.8

m

The object’s shadow is four times the post’s shadow. Its height is therefore four times

1.2

m, which is

4.8

m.

Key terms

Ratio: A comparison of two quantities by division.
Proportion: An equation stating that two ratios are equal.
Similar triangles: Triangles with the same shape, whose matching angles are equal and whose corresponding sides share a scale relationship.
Corresponding sides: Sides in different shapes that have matching positions or roles.
Scale relationship: The same multiplication factor connecting the lengths of corresponding sides.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M1. 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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