Linked triangles | Year 10

01 / LEARN

The idea, step by step

Use the shared side as a bridge

One triangle may contain enough information to find a height needed by another triangle. Label the shared side once and keep the same value in both calculations. Other sides need not be equal.

Choose a first triangle with two known sides, or a known side and acute angle. Then solve the second triangle.

A new height is not a drop

A sliding ladder has fixed length but changing height. Find both heights and subtract to calculate its drop. In adjacent triangles, the question may instead ask for the sum of two base lengths.

Write what each result measures and retain exact or full-precision values until the final operation.

A common trap

Giving the new height as the drop.

Instead: Drop means initial height minus final height; calculate both first.

02 / WORKED EXAMPLES

See the thinking

Example 1

Shared height

Two right triangles share a vertical height. The left has base 5 cm and hypotenuse 13 cm. The right base is 9 cm. Find the right hypotenuse and full base.

Shared height $= \sqrt{13^{2} - 5^{2}} = 12$ cm.
Right hypotenuse $= \sqrt{12^{2} + 9^{2}} = 15$ cm.
Full base $= 5 + 9 = 14$ cm.

15 cm; base 14 cm

Example 2

A change in height

A 13 m ladder moves from base distance 5 m to base distance 12 m. How far does its top drop?

First height $= \sqrt{13^{2} - 5^{2}} = 12$ m.
New height $= \sqrt{13^{2} - 12^{2}} = 5$ m.
Drop $= 12 - 5 = 7$ m.

7 m

Check your understanding

A right triangle has height 8 m and base 15 m. Find its hypotenuse.

Show answer

17 m

$\sqrt{8^{2} + 15^{2}} = 17$ m.

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Two right triangles share a height. One has base 8 cm and hypotenuse 17 cm; the other base is 20 cm. Find its hypotenuse.
3 marks
A 10 m ladder moves from base distance 6 m to 8 m. Find its top drop.
3 marks
A 12 m mast has ground anchors 5 m and 9 m from its base on opposite sides. Find both cable lengths and anchor separation.
3 marks
A symmetrical triangular roof has span 20 m and rise 9 m. Find either slope to 0.1 m.
3 marks
Points on level ground are 20 m and 40 m from a 12 m tower. Find both elevation angles to 0.1°.
3 marks
Two adjacent right triangles have bases 3 m and 4 m and share a height. Why cannot 7 m be used as the base of each triangle?
2 marks

Read the worked solutions

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

Two right triangles share a height. One has base 8 cm and hypotenuse 17 cm; the other base is 20 cm. Find its hypotenuse.
3 marks
Height $= \sqrt{17^{2} - 8^{2}} = 15$ cm.
Other hypotenuse $= \sqrt{15^{2} + 20^{2}} = 25$ cm.
25 cm
A 10 m ladder moves from base distance 6 m to 8 m. Find its top drop.
3 marks
Heights are $\sqrt{100 - 36} = 8$ m and $\sqrt{100 - 64} = 6$ m.
Drop $= 8 - 6 = 2$ m.
2 m
A 12 m mast has ground anchors 5 m and 9 m from its base on opposite sides. Find both cable lengths and anchor separation.
3 marks
Lengths are $\sqrt{12^{2} + 5^{2}} = 13$ m and $\sqrt{12^{2} + 9^{2}} = 15$ m.
Anchor separation $= 5 + 9 = 14$ m.
13 m and 15 m; 14 m apart
A symmetrical triangular roof has span 20 m and rise 9 m. Find either slope to 0.1 m.
3 marks
Half-span is 10 m.
Slope $= \sqrt{10^{2} + 9^{2}} = \sqrt{181}$ m.
13.5 m
Points on level ground are 20 m and 40 m from a 12 m tower. Find both elevation angles to 0.1°.
3 marks
Angles are ${tan}^{- 1} (12 / 20)$ and ${tan}^{- 1} (12 / 40)$ .
The nearer point has the larger angle.
31.0° and 16.7°
Two adjacent right triangles have bases 3 m and 4 m and share a height. Why cannot 7 m be used as the base of each triangle?
2 marks
Each calculation uses sides of one right triangle.
The bases remain 3 m and 4 m respectively.
7 m is the full span, not the base of either individual triangle.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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