Similarity and fixed trigonometric ratios | Year 9

01 / LEARN

The idea, step by step

Same shape, different size

Two right triangles with a matching acute angle have all their angles equal, so they are similar. Corresponding side lengths share one scale factor. Dividing one side by another cancels that factor.

For example, 3/4 and 9/12 both equal 0.75. Different-sized triangles can therefore have identical matching acute angles.

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

A ratio test is not a length test

Compare corresponding sides in the same units. The length units cancel, leaving a dimensionless ratio.

Scaling only one side while keeping another fixed generally changes the angle. Fixed trigonometric ratios apply to a fixed angle, not to all right triangles regardless of shape.

A common trap

Doubling the sides doubles each acute angle.

Instead: Uniform enlargement keeps angles fixed because corresponding side ratios stay equal.

02 / WORKED EXAMPLES

See the thinking

Example 1

Enlarge all three sides

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

The sides become $9, 12, 15$ .
$3 / 4 = 0.75$ and $9 / 12 = 0.75$ .
Its matching angle is unchanged.

Both ratios are 0.75

Example 2

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.

$a / 25 = 4 / 5$ .
$a = 25 (4 / 5) = 20$ .

20 cm

Check your understanding

Compare 6/8 and 18/24.

Show answer

Both are 0.75.

Multiplying both parts of a ratio by 3 preserves its value.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Compare opposite/adjacent for side pairs (3,4) and (9,12).
2 marks
Show that opposite/hypotenuse ratios 5/13 and 15/39 agree.
2 marks
Adjacent/hypotenuse = 4/5. Find adjacent when hypotenuse = 30 cm.
2 marks
Opposite/adjacent = 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 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 opposite/adjacent for side pairs (3,4) and (9,12).
2 marks
$9 / 12 = 3 / 4$ .
Matching angles agree.
Both are 3/4 = 0.75
Show that opposite/hypotenuse ratios 5/13 and 15/39 agree.
2 marks
$15 / 39 = (15 \div 3) / (39 \div 3) = 5 / 13$ .
Both are 5/13
Adjacent/hypotenuse = 4/5. Find adjacent when hypotenuse = 30 cm.
2 marks
$a = 30 (4 / 5) = 24$ cm.
24 cm
Opposite/adjacent = 3/4. Find opposite when adjacent = 20 m.
2 marks
$o / 20 = 3 / 4$ .
$o = 15$ 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 $6 / 3 = 2$ and $4 / 4 = 1$ .
There is no single uniform scale factor.
No
Three 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 opposite/adjacent ratio is 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 ↗

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