Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 3 - Similarity and trigonometric ratios

Similarity and trigonometric ratios

Use similarity to explain constant sine, cosine and tangent ratios and solve right-triangle problems.

Updated 2026-07-26 - 11 min read

Similarity and trigonometric ratios is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Corresponding sides in similar triangles

The orientation of a triangle can change without changing correspondence. Label vertices or angles first, then build ratios in the same order.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by corresponding sides in similar triangles and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Correspondence follows equal angles and opposite-side relationships, not page position.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Similar triangles have sides 4→10 and 6→x. Find x.

Step 1 - identify the governing idea: Equal corresponding angles establish similarity, and all matching side lengths share one scale factor.

Step 2 - apply it to this evidence: The scale factor is 10/4=2.5, so x=6×2.5.

Result: 15

The relationship is visible in the working: The scale factor is 10/4=2.5, so x=6×2.5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 12 — It does not agree with the required relationship: The scale factor is 10/4=2.5, so x=6×2.5.
  • 25 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A triangle with sides 3,4,5 is enlarged so its shortest side is 9. New hypotenuse?

Step 1 - identify the governing idea: Equal corresponding angles establish similarity, and all matching side lengths share one scale factor.

Step 2 - apply it to this evidence: Scale factor is 3, so 5×3=15.

Result: 15

The relationship is visible in the working: Scale factor is 3, so 5×3=15. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 11 — It does not agree with the required relationship: Scale factor is 3, so 5×3=15.
  • 45 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Two triangles have angles 35°,55°,90°. What relationship follows?

Step 1 - identify the governing idea: Equal corresponding angles establish similarity, and all matching side lengths share one scale factor.

Step 2 - apply it to this evidence: AAA equality fixes shape though not size.

Result: They are similar

The relationship is visible in the working: AAA equality fixes shape though not size. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • They are congruent — It does not agree with the required relationship: AAA equality fixes shape though not size.
  • They have equal areas — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No relationship follows — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Constant right-triangle ratios

Sine, cosine and tangent name these invariant ratios. Triangle size changes every side by the same factor, which cancels in a ratio.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by constant right-triangle ratios and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The lengths can change, but corresponding ratios remain constant.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

A 30° right triangle is doubled in size. What happens to opposite/hypotenuse?

Step 1 - identify the governing idea: All right triangles sharing an acute angle are similar by AA, so corresponding side ratios are invariant.

Step 2 - apply it to this evidence: Both lengths double, so the factor cancels.

Result: It stays the same

The relationship is visible in the working: Both lengths double, so the factor cancels. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • It doubles — It does not agree with the required relationship: Both lengths double, so the factor cancels.
  • It halves — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It becomes 30 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Why do two right triangles with a shared 40° angle have equal tangent?

Step 1 - identify the governing idea: All right triangles sharing an acute angle are similar by AA, so corresponding side ratios are invariant.

Step 2 - apply it to this evidence: AA similarity preserves corresponding ratios.

Result: They are similar, so opposite/adjacent is constant

The relationship is visible in the working: AA similarity preserves corresponding ratios. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Their areas are equal — It does not agree with the required relationship: AA similarity preserves corresponding ratios.
  • All right triangles are congruent — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Tangent is the angle in degrees — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

For a fixed acute angle, which ratio is cosine?

Step 1 - identify the governing idea: All right triangles sharing an acute angle are similar by AA, so corresponding side ratios are invariant.

Step 2 - apply it to this evidence: Cosine compares adjacent with hypotenuse.

Result: adjacent/hypotenuse

The relationship is visible in the working: Cosine compares adjacent with hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • opposite/hypotenuse — It does not agree with the required relationship: Cosine compares adjacent with hypotenuse.
  • opposite/adjacent — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • hypotenuse/opposite — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Choose and interpret a trig ratio

Opposite and adjacent swap when the reference angle changes, while the hypotenuse remains opposite the right angle.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by choose and interpret a trig ratio and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Relabel them for the specific acute angle named in the question.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

θ=45°, adjacent=7. Find opposite.

Step 1 - identify the governing idea: Choose sine, cosine or tangent from the side pair relative to the named angle, write the equation, solve and interpret with units and accuracy.

Step 2 - apply it to this evidence: tan45°=opposite/7=1.

Result: 7

The relationship is visible in the working: tan45°=opposite/7=1. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 4.95 — It does not agree with the required relationship: tan45°=opposite/7=1.
  • 14 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 49 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Opposite=5, hypotenuse=13. Which equation finds θ?

Step 1 - identify the governing idea: Choose sine, cosine or tangent from the side pair relative to the named angle, write the equation, solve and interpret with units and accuracy.

Step 2 - apply it to this evidence: Sine uses opposite over hypotenuse.

Result: sinθ=5/13

The relationship is visible in the working: Sine uses opposite over hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • cosθ=5/13 — It does not agree with the required relationship: Sine uses opposite over hypotenuse.
  • tanθ=13/5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • sinθ=13/5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A ramp rises 1.2 m over horizontal run 8 m. Angle to nearest degree?

Step 1 - identify the governing idea: Choose sine, cosine or tangent from the side pair relative to the named angle, write the equation, solve and interpret with units and accuracy.

Step 2 - apply it to this evidence: tanθ=1.2/8=0.15, so θ≈8.53°.

Result: 9°

The relationship is visible in the working: tanθ=1.2/8=0.15, so θ≈8.53°. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 81° — It does not agree with the required relationship: tanθ=1.2/8=0.15, so θ≈8.53°.
  • 7° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.15° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. A triangle with sides 3,4,5 is enlarged so its shortest side is 9. New hypotenuse?
  2. Why do two right triangles with a shared 40° angle have equal tangent?
  3. Opposite=5, hypotenuse=13. Which equation finds θ?

Answers

  1. 15 — Scale factor is 3, so 5×3=15.
  2. They are similar, so opposite/adjacent is constant — AA similarity preserves corresponding ratios.
  3. sinθ=5/13 — Sine uses opposite over hypotenuse.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources