Find corresponding side lengths in similar triangles

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Match each side, keep one direction, and use a scale factor

Suppose a triangle is enlarged without changing its shape. Each side becomes longer by the same multiplier. The enlarged triangle is similar to the original triangle. If one side changes from

4

cm to

10

cm, the multiplier is

2.5

. A matching side of

6

cm would then become

15

cm. To solve these questions reliably, first match the sides. Then use the same scale factor for each matching pair.

What you will learn

Explain what it means for two triangles to be similar.
Use the order of triangle names to identify corresponding sides.
Find a missing side length using a scale factor or an equal-ratio equation.
Check that an answer makes sense for the direction of the scale change.

1. Grade 9 bridge: ratios and scale factors

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

10

to

4

is

\frac{10}{4}

, or

2.5

. Ratios help us compare matching side lengths.

A scale factor is the multiplier that changes one length into its matching length in another triangle. To find it, divide a length in the triangle you are moving to by its matching length in the triangle you are starting from. From a length of

4

to a matching length of

10

, the scale factor is

\frac{10}{4}=2.5

.

The direction matters. From a smaller triangle to a larger triangle, the scale factor is greater than

1

. From a larger triangle to a smaller triangle, it is less than

1

. For the same pair of lengths, the reverse scale factor is

\frac{4}{10}=0.4

.

You can undo multiplication with division. For example, if

2.5x=10

, divide both sides by

2.5

to find

x=4

. This is useful when you know the scale factor and the larger length, but need the matching smaller length.

In words, the scale factor is the target-triangle length divided by the matching starting-triangle length.

k=\frac{\text{target length}}{\text{matching starting length}}

A scale factor changes a side length into its matching length.
Name the starting triangle and target triangle before calculating.
Use division to undo multiplication when the unknown length is the starting length.

2. Similar triangles: identify matching sides

Similar triangles have the same shape. Their corresponding angles are equal, and their corresponding side lengths are proportional. Corresponding means matching parts that occupy the same position in the figures. Proportional means that ratios of matching lengths are equal.

The order of the letters in a similarity statement tells you which vertices correspond. A vertex is a corner of a triangle. If

\triangle ABC\sim\triangle DEF

, then

A

matches

D

,

B

matches

E

, and

C

matches

F

. The symbol

\sim

means “is similar to.”

The vertex matches show that side

AB

corresponds to

DE

, side

BC

corresponds to

EF

, and side

AC

corresponds to

DF

. Read the letters in order. Do not choose pairs just because the sides look alike in a drawing; a diagram may not be drawn to scale.

A proportion is an equation that says two ratios are equal. For similar triangles, you can compare matching side lengths using a proportion. Keep the comparison direction consistent: if the first ratio has a larger-triangle length over a smaller-triangle length, use that order in the other ratio too.

Before calculating, write down which sides correspond. Correct side matching is necessary for the scale factor or proportion to give the right answer.

\triangle ABC\sim\triangle DEF\Rightarrow AB\leftrightarrow DE,\ BC\leftrightarrow EF,\ AC\leftrightarrow DF

Similar triangles have equal corresponding angles and proportional corresponding side lengths.
The order of the triangle names identifies matching vertices and sides.
Use the same direction in every ratio.

3. Organize side pairs before calculating

A table can keep matching sides together. Suppose the sides of a smaller triangle are

4

,

6

, and

8

. The corresponding sides in a larger similar triangle are

10

,

15

, and an unknown length. The first two pairs both give a scale factor of

2.5

from smaller to larger.

That same multiplier must apply to the remaining pair. The table shows how to organize the information. It does not prove that the triangles are similar; the question must state or otherwise provide that they are similar and show which sides match.

If known matching pairs do not give the same scale factor, check the side matches and calculations. Similar triangles use one consistent scale factor for all corresponding sides.

\frac{10}{4}=\frac{15}{6}=2.5

Place each side beside its corresponding side.
Known matching pairs should give the same scale factor.
Apply that scale factor to the matching side with the unknown length.

4. Guided example and independent practice

For a missing side, begin with the vertex matches. Use a known pair to find the scale factor in the direction you need. Apply it to the corresponding side, then check that the answer size fits the direction of the change.

Try these questions after studying the worked example. First, matching sides are

5

cm in the smaller triangle and

12

cm in the larger triangle. Another smaller side is

7

cm. Find its matching larger length. Second, a larger side of

18

m matches a smaller side of

12

m. Another larger side is

21

m. Find its matching smaller length. State the direction of your scale factor in each question.

Include the units from the question. A useful final check is to ask whether the answer should be larger or smaller than the side it matches. An enlargement uses a factor greater than

1

; a reduction uses a factor less than

1

.

Match sides before finding a scale factor.
Use the factor on the side that corresponds to the unknown.
Check the direction, answer size, and units.

Matching sides and scale factor

Smaller triangle side	Matching larger triangle side	Multiplier from smaller to larger
$4$	$10$	$2.5$
$6$	$15$	$2.5$
$8$	unknown	use $2.5$

Worked example

Find a missing side in the larger triangle

Triangle

PQR

is similar to triangle

XYZ

. The vertex order means

P

matches

X

,

Q

matches

Y

, and

R

matches

Z

. In the smaller triangle,

PQ=6

cm and

QR=9

cm. In the larger triangle, the matching side

XY=10

cm. Find

YZ

.

Match the sides
The vertex matches show that side $PQ$ corresponds to $XY$ , and side $QR$ corresponds to $YZ$ . The known matching lengths are $6$ cm and $10$ cm.
$PQ\leftrightarrow XY,\quad QR\leftrightarrow YZ$
Find the scale factor
We are moving from the smaller triangle to the larger triangle. Divide the larger known length by the smaller matching length. This gives the multiplier for every corresponding side in that direction.
$k=\frac{10}{6}=\frac{5}{3}$
Calculate the missing length
Side $QR$ is $9$ cm, and its matching side is $YZ$ . Multiply $9$ cm by the scale factor $\frac{5}{3}$ to find the larger length. YZ=9× $\frac{5}{3}$ =15
Check the result
The scale factor is greater than $1$ , so the matching side in the larger triangle should be longer than $9$ cm. The result of $15$ cm fits that expectation.

Answer:

YZ=15

cm.

Check: The matching-side ratios agree:

\frac{10}{6}=\frac{15}{9}=\frac{5}{3}

. The answer is in centimetres and is longer than its matching side in the smaller triangle.

Common mistakes and how to avoid them

Matching sides by how long they look in a drawing.

Correction: Use the vertex order in the similarity statement or the matching information provided. The drawing may not be to scale.

Using larger over smaller in one ratio and smaller over larger in another.

Correction: Choose a direction and keep it the same in every ratio used to find the scale factor.

Applying the scale factor to a side that does not match the unknown.

Correction: List the corresponding side pairs first. Use the factor on the side paired with the unknown.

Giving a shorter length for a side in the enlarged triangle.

Correction: Check the direction. A scale factor greater than

1

makes each matching length larger.

Lesson summary

Similar triangles have the same shape and proportional corresponding side lengths.
Use the order of the triangle names to identify matching vertices and sides.
Find a scale factor from a known pair of corresponding sides.
Apply the same scale factor to the missing corresponding side.
Check the direction, answer size, and units.

Check your understanding

Question 1

Two similar triangles have corresponding sides of

8

cm in the smaller triangle and

12

cm in the larger triangle. A second smaller side is

10

cm. What is its matching larger side?

$15$ cm
$6.7$ cm
$14$ cm
$20$ cm

Show answer and explanation

15

cm

The scale factor from smaller to larger is

\frac{12}{8}=1.5

. Multiply the matching smaller side by

1.5

:

10\times1.5=15

cm.

Question 2

A larger triangle has a side of

20

m matching a side of

12

m in a smaller triangle. What is the scale factor from the larger triangle to the smaller triangle?

$\frac{5}{3}$
$\frac{3}{5}$
$8$
$32$

Show answer and explanation

\frac{3}{5}

The direction is from larger to smaller, so divide the smaller matching length by the larger length:

\frac{12}{20}=\frac{3}{5}

.

Question 3

If

\triangle JKL\sim\triangle MNO

, which side corresponds to

KL

?

$MN$
$MO$
$NO$
$JK$

Show answer and explanation

NO

The vertex order gives

J

to

M

,

K

to

N

, and

L

to

O

. Therefore, side

KL

corresponds to side

NO

.

Key terms

Similar triangles: Triangles with the same shape, equal corresponding angles, and side lengths in the same ratio.
Corresponding: Matching parts that occupy the same position in two figures.
Scale factor: The multiplier that changes a length into its matching length in another figure.
Ratio: A comparison of two quantities by division.
Proportion: An equation that shows two ratios are equal.

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