Selecting the right method | Year 9

01 / LEARN

The idea, step by step

Let the givens choose the method

Two side lengths and a missing side suggest Pythagoras. An acute angle and one side suggest a trigonometric ratio for another side. Two sides and a missing acute angle suggest an inverse ratio.

These methods require a right triangle in this chapter. In a practical situation, identify a perpendicular height or draw a justified right triangle first.

Enough information, not more calculation

Angles fix a triangle’s shape but not its size. A right triangle with a 45° acute angle could have legs of length 1, 10 or 100. An actual length needs a side measurement or scale condition.

A valid method used for the wrong quantity still gives a wrong response. Final height and a drop in height are different quantities. Check the wording, units and diagram before finishing.

A common trap

Use Pythagoras whenever the picture contains a triangle.

Instead: It requires a right triangle and two side lengths. One angle and one side usually call for trigonometry.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two known sides

A right triangle has legs 9 m and 12 m. Find the hypotenuse.

Two sides are known and the unknown is a side: use Pythagoras.
$c = \sqrt{9^{2} + 12^{2}} = 15$ m.

15 m

Example 2

One angle and one side

Adjacent = 8 cm and θ = 53°. Find opposite.

Opposite and adjacent select tangent.
$x = 8 tan 53^{\circ} \approx 10.6164 \dots$ cm.

10.62 cm

Check your understanding

You know hypotenuse and opposite and need the angle. Which method?

Show answer

Inverse sine

$θ = {sin}^{- 1} (o / h)$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Legs 7 and 10 are known. Which method finds the third side? Write its equation.
2 marks
Hypotenuse and an acute angle are known. Which equation finds adjacent?
2 marks
Opposite and hypotenuse are known. Write the expression for the acute angle.
2 marks
A right triangle has one acute angle 45° but no side length. Can you find its hypotenuse? Explain.
3 marks
A student uses c = a + b for legs 3 and 4. Correct the calculation.
2 marks
A right triangle has legs 3 cm and 4 cm. Find the hypotenuse, then the acute angle opposite 3 cm to two decimal places.
4 marks

Read the worked solutions

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

Legs 7 and 10 are known. Which method finds the third side? Write its equation.
2 marks
Two sides are known; a side is missing.
Pythagoras: $c^{2} = 7^{2} + 10^{2}$
Hypotenuse and an acute angle are known. Which equation finds adjacent?
2 marks
Cosine connects adjacent and hypotenuse.
$a = h cos θ$
Opposite and hypotenuse are known. Write the expression for the acute angle.
2 marks
Calculate opposite/hypotenuse, then use inverse sine.
$θ = {sin}^{- 1} (o / h)$
A right triangle has one acute angle 45° but no side length. Can you find its hypotenuse? Explain.
3 marks
Angles fix shape, not scale.
Many similar triangles satisfy the angle condition.
No
A student uses c = a + b for legs 3 and 4. Correct the calculation.
2 marks
$c^{2} = 3^{2} + 4^{2} = 25$ .
$c = 5$ , not 7.
c = 5
A right triangle has legs 3 cm and 4 cm. Find the hypotenuse, then the acute angle opposite 3 cm to two decimal places.
4 marks
$c = \sqrt{9 + 16} = 5$ cm.
$θ = {tan}^{- 1} (3 / 4) \approx {36.87}^{\circ}$ .
5 cm; 36.87°

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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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