T4 Solve realistic problems with similar triangles

Ontario Grade 10 Mathematics

Trigonometry

Use matching shapes and side-length ratios to find an unknown measurement

A triangle can help measure something that is too tall or too far away to measure directly. For example, a tree and its shadow form a triangle, while a person and their shadow form another. If the sunlight reaches both at the same angle, the triangles have the same shape. Their side lengths can then be compared. This lesson reviews ratios and proportions, explains how to recognize similar triangles, and shows how to use them to solve a measurement problem.

What you will learn

Explain what makes two triangles similar.
Identify corresponding sides in a realistic situation.
Set up and solve a proportion to find an unknown length.
Check whether an answer makes sense in the context.

1. Grade 9 bridge: ratios and proportions

A ratio compares two quantities by division. For example, a drawing that is 4 cm wide and 2 cm tall has a width-to-height ratio of 2 to 1. A proportion is a statement that two ratios are equal. Proportions are useful when two shapes have the same shape but different sizes.

To solve a proportion, you can use equivalent ratios or cross-multiply. Cross-multiplying means multiplying the numerator of one ratio by the denominator of the other. It helps find an unknown value, but first make sure the ratios compare matching measurements in the same order.

For example, if one small shape has a side of 3 and its matching side on a larger shape is 9, the scale factor from small to large is 3. A matching small side of 5 would correspond to a large side of 15. A scale factor tells how many times larger or smaller corresponding lengths are.

\frac{a}{b}=\frac{c}{d}

Keep the order of corresponding measurements consistent.
A proportion states that two ratios are equal.
A scale factor compares the lengths of matching sides.

2. What similar triangles tell us

Similar triangles have the same shape, but they may have different sizes. Their matching angles are equal, and their corresponding sides are in the same ratio. Corresponding means that the sides occupy matching positions in the two triangles.

In realistic measurement problems, a diagram or the situation may show why the triangles are similar. For a shadow problem, each object is vertical and meets level ground, so each triangle has a right angle. The sunlight reaches both objects at the same angle during the same moment. Those matching angles make the triangles the same shape.

A diagram does not need to be drawn to scale. Use the information in the problem to identify which sides match. In a shadow situation, the object's height matches the other object's height, and each shadow matches the other shadow. Do not match sides just because they look close together in a sketch.

Once the matching sides are identified, write a proportion. Put the first triangle's corresponding height and shadow in one ratio, and the second triangle's height and shadow in the other. Keep both ratios in the same order. If one height is unknown, the proportion can be solved for it.

\frac{\text{height}_1}{\text{shadow}_1}=\frac{\text{height}_2}{\text{shadow}_2}

Similar triangles have equal matching angles and proportional matching sides.
Corresponding sides must be paired by their roles in the triangles.
In a proportion, compare height with height and shadow with shadow.

3. A reliable plan for realistic problems

Start by naming the two triangles in the situation. State why they have the same shape, using information such as vertical objects, level ground, and sunlight reaching both at the same angle. Do not assume that two triangles are similar only because they appear similar in a drawing.

Next, label each known length and the unknown length. Include units such as metres or centimetres. Identify matching sides before writing any equation. If the unknown is a height, pair it with the known height; if the unknown is a shadow, pair it with the known shadow.

Write a proportion with corresponding sides in matching order. Substitute the known values, then solve for the unknown. When multiplying or dividing, keep the units in mind. Finally, decide whether the result is reasonable. A tree should be taller than a person, for instance, if its shadow is several times longer under the same sunlight.

If the problem gives measurements in different units, convert them first so that the lengths being compared use the same unit. For example, do not compare a height in centimetres with a shadow in metres without converting one of them.

Explain why the triangles have the same shape before using a proportion.
Label units and match sides by their meaning.
Check that the answer fits the situation and has an appropriate unit.

4. Independent practice and a final check

Try a similar problem on your own: a sign that is 1.2 m tall casts a shadow that is 0.9 m long. At the same time, a nearby building casts a shadow that is 6.3 m long. Find the building's height. First identify the corresponding heights and shadows, then write a proportion. Keep the units consistent and explain briefly why the triangles have the same shape.

Before accepting an answer, ask yourself three questions. Did I use matching sides? Did I keep the same order in both ratios? Does the size of my answer make sense compared with the known measurements? These checks can catch a reversed ratio or an arithmetic error.

Practise the setup before using a calculator.
A longer shadow under the same sunlight corresponds to a proportionally taller object.
Check both the arithmetic and the real-world meaning.

Matching measurements in a shadow problem

Triangle	Height	Shadow
Person	1.6 m	2.1 m
Tree	Unknown	8.4 m
Matching sides	Height ↔ height	Shadow ↔ shadow

Worked example

Finding a tree's height from shadows

At the same moment, a person who is 1.6 m tall casts a shadow that is 2.1 m long. A nearby tree casts a shadow that is 8.4 m long. How tall is the tree?

Identify the triangles
The person's height and shadow form one right triangle, and the tree's height and shadow form another. Each object is vertical and the ground is level, so each triangle has a right angle. The sunlight reaches both during the same moment, so the light makes the same angle with the ground. The triangles are similar.
Match the sides
The person's height corresponds to the tree's height. The person's shadow corresponds to the tree's shadow. Let $h$ represent the tree's height in metres. Compare height with shadow in both ratios.
$\frac{1.6}{2.1}=\frac{h}{8.4}$
Solve the proportion
Cross-multiply to isolate the unknown height. Divide by 2.1 to find the value of $h$ .
$h=\frac{1.6\times 8.4}{2.1}=6.4$
State and check the result
The tree is 6.4 m tall. Its shadow is four times the person's shadow, and the calculated height is also four times the person's height. That consistent scale factor supports the result.
$\frac{8.4}{2.1}=\frac{6.4}{1.6}=4$

Answer: The tree is 6.4 m tall.

Check: The tree's shadow is four times as long as the person's shadow, so its height should be four times 1.6 m. That gives 6.4 m, which matches the proportion.

Common mistakes and how to avoid them

Comparing the person's height with the tree's shadow.

Correction: Match sides that play the same role: height with height and shadow with shadow.

Writing the first ratio as height over shadow and the second as shadow over height.

Correction: Keep the order consistent in both ratios. For example, use height over shadow for each triangle.

Assuming triangles are similar because they look alike in a sketch.

Correction: Use facts from the situation, such as matching right angles and the same sunlight angle, to explain why the triangles have the same shape.

Giving a number without a unit or without checking whether it is reasonable.

Correction: Include the appropriate length unit and compare the result with the known measurements.

Lesson summary

Similar triangles have the same shape, equal matching angles, and proportional corresponding sides.
In realistic problems, identify why the triangles are similar and match sides by their roles.
Write a proportion in a consistent order, solve for the unknown, and check the result in context.

Check your understanding

Question 1

A 1.5 m post casts a 2 m shadow. At the same time, a tree casts a 10 m shadow. Which proportion correctly represents the tree's height,

h

?

$\frac{1.5}{2}=\frac{h}{10}$
$\frac{1.5}{2}=\frac{10}{h}$
$\frac{2}{1.5}=\frac{h}{10}$
correctIndex`: 0, «analysis»: «Need correct JSON.»

Show answer and explanation

\frac{1.5}{2}=\frac{h}{10}

The post's height corresponds to the tree's height, and the post's shadow corresponds to the tree's shadow. Comparing height over shadow for both gives the first proportion. It leads to a tree height of 7.5 m, which is greater than the post's height as expected.

Question 2

In the same setup, which is the tree's height?

5 m
7.5 m
13.3 m
correctIndex` : 1,

Show answer and explanation

7.5 m

The tree's shadow is five times the post's shadow. The similar triangle has the same scale factor for height, so the tree's height is five times 1.5 m, or 7.5 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, equal matching angles, and corresponding side lengths in the same ratio.
Corresponding sides: Sides in matching positions or with matching roles in two shapes.
Scale factor: The number that tells how a length changes from one similar shape to another.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T4. It is a study resource, not an official curriculum publication.

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T4 · Solve realistic problems with similar triangles