Similarity and fixed trigonometric ratios | Year 9

01 / LEARN

Understand the idea.

Same shape, different size

Right triangles sharing an acute angle also share the other acute angle, because their angles total $180^{\circ}$ . They are therefore similar. Their corresponding lengths differ by one scale factor, while the matching angles remain unchanged.

A common scale factor cancels in a side ratio: $\frac{3 k}{4 k} = \frac{3}{4}$ for $k > 0$ . This explains why a trigonometric ratio depends on the chosen angle rather than the overall size of the triangle.

\frac{k o}{k a} = \frac{o}{a}

A ratio test is not a length test

Compare corresponding sides using matching units before dividing. The units cancel, leaving a dimensionless number. For a fixed acute angle, opposite-to-adjacent and adjacent-to-hypotenuse ratios stay constant across similar right triangles.

Increasing only one leg is not a uniform enlargement. It generally changes the shape and the acute angles. To preserve an angle, every corresponding side must scale together; simply making a triangle larger in one direction is insufficient.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Enlarge all three sides

Enlarge a 3–4–5 triangle by a factor of 3 and compare opposite/adjacent.

Worked solution

1
Multiply every side by the scale factor.
$\begin{aligned} 3 \times 3 & = 9 \\ 4 \times 3 & = 12 \\ 5 \times 3 & = 15 \end{aligned}$
A uniform enlargement scales all three corresponding lengths equally.
2
Calculate the original opposite-to-adjacent ratio.
$\frac{o}{a} = \frac{3}{4} = 0.75$
The chosen angle pairs the three-unit opposite side with the four-unit adjacent side.
3
Calculate the image ratio.
$\frac{o^{'}}{a^{'}} = \frac{9}{12} = \frac{3}{4} = 0.75$
The common factor of three cancels, leaving the same ratio and matching angle.

Answer

Both ratios are 0.75

EXAMPLE 02

Recover a corresponding length

Adjacent/hypotenuse is 4/5 for a chosen acute angle. A similar right triangle has hypotenuse 25 cm. Find its corresponding adjacent side.

Worked solution

1
Write the fixed ratio for the new triangle.
$\frac{a}{25} = \frac{4}{5}$
Similarity preserves the adjacent-to-hypotenuse ratio for the chosen angle.
2
Multiply by the hypotenuse length.
$a = 25 \times \frac{4}{5} = 20 cm$
Multiplying both sides by twenty-five isolates the adjacent length.
3
Check the ratio and side size.
$\frac{20}{25} = \frac{4}{5}$
The required ratio is recovered, and the adjacent side is shorter than the hypotenuse.

Answer

20 cm

03 / YOUR TURN

Try it yourself.

Check your understanding

Compare 6/8 and 18/24.

Show answer & reasoning

Both are 0.75.

$\frac{6}{8} = \frac{3}{4} = 0.75$ and $\frac{18}{24} = \frac{3}{4} = 0.75$ . Multiplying both numerator and denominator by three changes the lengths but preserves their ratio.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Compare $\frac{opposite}{adjacent}$ for side pairs $(3, 4)$ and $(9, 12)$ .
2 marks
Show that $\frac{opposite}{hypotenuse}$ ratios $\frac{5}{13}$ and $\frac{15}{39}$ agree.
2 marks
$\frac{Adjacent}{hypotenuse} = \frac{4}{5}$ . Find adjacent when hypotenuse $= 30$ cm.
2 marks
$\frac{Opposite}{adjacent} = \frac{3}{4}$ . Find opposite when adjacent $= 20$ m.
2 marks
Corresponding legs are 3,4 in one triangle and 6,4 in another. Are they similar?
3 marks
Three $\frac{opposite}{adjacent}$ pairs are $(6, 8)$ , $(12, 16)$ and $(18, 24)$ . State the common ratio and explain.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Compare $\frac{opposite}{adjacent}$ for side pairs $(3, 4)$ and $(9, 12)$ .
2 marks
$\frac{9}{12} = \frac{3}{4}$ .
Matching angles agree.
Both are $\frac{3}{4} = 0.75$
Show that $\frac{opposite}{hypotenuse}$ ratios $\frac{5}{13}$ and $\frac{15}{39}$ agree.
2 marks
$\frac{15}{39} = \frac{15 \div 3}{39 \div 3} = \frac{5}{13}$ .
Both are $\frac{5}{13}$
$\frac{Adjacent}{hypotenuse} = \frac{4}{5}$ . Find adjacent when hypotenuse $= 30$ cm.
2 marks
$a = 30 (\frac{4}{5}) = 24$ cm.
24 cm
$\frac{Opposite}{adjacent} = \frac{3}{4}$ . Find opposite when adjacent $= 20$ m.
2 marks
$\frac{o}{20} = \frac{3}{4}$ .
$o = 15$ m.
15 m
Corresponding legs are 3,4 in one triangle and 6,4 in another. Are they similar?
3 marks
The scale factors are $\frac{6}{3} = 2$ and $\frac{4}{4} = 1$ .
There is no single uniform scale factor.
No
Three $\frac{opposite}{adjacent}$ pairs are $(6, 8)$ , $(12, 16)$ and $(18, 24)$ . State the common ratio and explain.
3 marks
All pairs are uniform enlargements of $(3, 4)$ .
Each $\frac{opposite}{adjacent}$ ratio is $\frac{3}{4}$ .
0.75

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Mathematics

Year 9 similarity, Pythagoras and right-triangle trigonometric ratios, with prerequisite revision. Non-right-triangle sine/cosine rules are not required.

Official curriculum source ↗

In this lesson

Understand the idea.

Same shape, different size

A ratio test is not a length test

See the method in action.

Enlarge all three sides

Multiply every side by the scale factor.

Calculate the original opposite-to-adjacent ratio.

Calculate the image ratio.

Recover a corresponding length

Write the fixed ratio for the new triangle.

Multiply by the hypotenuse length.

Check the ratio and side size.

Try it yourself.

Check your understanding

Keep practising