Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 3 - Similarity, Pythagoras and right-triangle trigonometry

Similarity, Pythagoras and right-triangle trigonometry

Solve spatial problems using similarity, Pythagoras and sine, cosine or tangent ratios.

Updated 2026-07-26 - 10 min read

Similarity, Pythagoras and right-triangle trigonometry 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.

Use similarity and scale

A scale factor must compare matching sides in a consistent direction. A diagram's orientation does not determine correspondence.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by use similarity and scale 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: Use angle or vertex matching, then compare the associated opposite sides.

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

Triangles have corresponding sides 3→9 and 5→x. Find x.

Step 1 - identify the governing idea: Similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: The scale factor is 3, so x=5×3.

Result: 15

The relationship is visible in the working: The scale factor is 3, so x=5×3. 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: The scale factor is 3, so x=5×3.
  • 45 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5/3 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A 1.5 m person casts a 2 m shadow; a tree casts an 8 m shadow. Tree height?

Step 1 - identify the governing idea: Similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: Similar triangles give h/8=1.5/2.

Result: 6 m

The relationship is visible in the working: Similar triangles give h/8=1.5/2. 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 m — It does not agree with the required relationship: Similar triangles give h/8=1.5/2.
  • 5.5 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 16/1.5 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Two triangles share equal angles but one is rotated. Are they similar?

Step 1 - identify the governing idea: Similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: Rotation does not change angle or side ratios.

Result: Yes, equal corresponding angles establish similarity

The relationship is visible in the working: Rotation does not change angle or side 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:

  • No, orientation must match — It does not agree with the required relationship: Rotation does not change angle or side ratios.
  • Only if side lengths are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only if areas are equal — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Apply Pythagoras

Identify the right angle first, label the opposite side c, substitute lengths with units and take the positive square root for a distance.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by apply pythagoras 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 hypotenuse is defined by being opposite the right angle, not by drawing scale.

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

Right triangle legs are 9 and 12. Hypotenuse?

Step 1 - identify the governing idea: Pythagoras relates squared side lengths in a right triangle. Rearrange by subtracting when finding a shorter side.

Step 2 - apply it to this evidence: c=√(81+144)=√225.

Result: 15

The relationship is visible in the working: c=√(81+144)=√225. 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:

  • 21 — It does not agree with the required relationship: c=√(81+144)=√225.
  • √3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 225 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Hypotenuse is 13 and one leg is 5. Other leg?

Step 1 - identify the governing idea: Pythagoras relates squared side lengths in a right triangle. Rearrange by subtracting when finding a shorter side.

Step 2 - apply it to this evidence: b=√(13²−5²)=√144.

Result: 12

The relationship is visible in the working: b=√(13²−5²)=√144. 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:

  • 18 — It does not agree with the required relationship: b=√(13²−5²)=√144.
  • √194 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 14 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Can Pythagoras be used directly on a non-right triangle?

Step 1 - identify the governing idea: Pythagoras relates squared side lengths in a right triangle. Rearrange by subtracting when finding a shorter side.

Step 2 - apply it to this evidence: The theorem's condition is a right angle.

Result: No, not without first creating or proving a right triangle

The relationship is visible in the working: The theorem's condition is a right angle. 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:

  • Yes, for every triangle — It does not agree with the required relationship: The theorem's condition is a right angle.
  • Only when all sides are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only for obtuse triangles — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Choose and use trigonometric ratios

Opposite and adjacent depend on the selected acute angle; the hypotenuse does not. Write the ratio before substituting.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by choose and use trigonometric 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: Relabel opposite and adjacent relative to the stated angle.

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

In a right triangle, θ=30°, hypotenuse=10. Find opposite.

Step 1 - identify the governing idea: Relative to angle θ: sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, tanθ=opposite/adjacent.

Step 2 - apply it to this evidence: sin30°=opposite/10, so opposite=5.

Result: 5

The relationship is visible in the working: sin30°=opposite/10, so opposite=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:

  • 8.66 — It does not agree with the required relationship: sin30°=opposite/10, so opposite=5.
  • 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.05 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Adjacent=8, opposite=6. Find θ to nearest degree.

Step 1 - identify the governing idea: Relative to angle θ: sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, tanθ=opposite/adjacent.

Step 2 - apply it to this evidence: tanθ=6/8, so θ=tan⁻¹(0.75)≈36.9°.

Result: 37°

The relationship is visible in the working: tanθ=6/8, so θ=tan⁻¹(0.75)≈36.9°. 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:

  • 53° — It does not agree with the required relationship: tanθ=6/8, so θ=tan⁻¹(0.75)≈36.9°.
  • 48° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A ladder length 5 m reaches 4 m up a wall. Find its angle with ground.

Step 1 - identify the governing idea: Relative to angle θ: sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, tanθ=opposite/adjacent.

Step 2 - apply it to this evidence: sinθ=4/5 or cosθ=3/5 gives θ≈53.1°.

Result: Approximately 53°

The relationship is visible in the working: sinθ=4/5 or cosθ=3/5 gives θ≈53.1°. 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:

  • 37° — It does not agree with the required relationship: sinθ=4/5 or cosθ=3/5 gives θ≈53.1°.
  • 80° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4/5° — 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 1.5 m person casts a 2 m shadow; a tree casts an 8 m shadow. Tree height?
  2. Hypotenuse is 13 and one leg is 5. Other leg?
  3. Adjacent=8, opposite=6. Find θ to nearest degree.

Answers

  1. 6 m — Similar triangles give h/8=1.5/2.
  2. 12 — b=√(13²−5²)=√144.
  3. 37° — tanθ=6/8, so θ=tan⁻¹(0.75)≈36.9°.

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