Practical length problems | Year 8

01 / LEARN

The idea, step by step

Draw the model and identify the requested quantity

A wall perpendicular to level ground creates a right angle. A taut ladder or cable can be a hypotenuse. A rectangle diagonal also divides the rectangle into right triangles.

Label known lengths and the unknown on your own diagram. Decide whether you are finding a longest side or a shorter side before choosing addition or subtraction.

Interpret the mathematical result

An answer might be a height, a diagonal or a change between two heights. Read the wording after calculating: the new height of a sliding ladder is not itself the distance the top moved.

Check practical assumptions such as a level surface, perpendicular wall or straight cable. State units and choose accuracy suited to the measurements.

A common trap

Answering a requested drop with the final height.

Instead: Calculate both heights and subtract. Label the quantity being requested before working.

02 / WORKED EXAMPLES

See the thinking

Example 1

A rectangular diagonal

A rectangular field is 30 m by 40 m. Find the straight distance between opposite corners.

The sides and diagonal form a right triangle.
$d^{2} = 30^{2} + 40^{2} = 2500$ .
Take the root and check that 50 is between 40 and 70.

50 m.

Example 2

A drop is a difference

A 10 m ladder has its foot 6 m from a wall, then 8 m from the wall. How far does its top move down?

Initially the height is $\sqrt{100 - 36} = 8$ m.
Finally it is $\sqrt{100 - 64} = 6$ m.
Subtract the final height from the initial height: 8 - 6 = 2 m.

2 m.

Check your understanding

A 13 m ladder has its foot 5 m from a perpendicular wall. How high does it reach?

Show answer

12 m.

$h^{2} = 169 - 25 = 144$ , so h = 12.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

A rectangular screen is 12 cm by 16 cm. Find its diagonal.
2 marks
A 17 m ladder has its foot 8 m from a perpendicular wall. Find the height reached.
2 marks
A cable runs from the top of a 12 m pole to a point 5 m horizontally from its base. Find the taut cable length.
2 marks
A 25 m wire reaches a pole top 24 m above level ground. Find the horizontal distance of its ground anchor from the pole.
2 marks
A 15 m ladder moves from a foot distance of 9 m to a foot distance of 12 m from a wall. Find the vertical drop of its top.
3 marks
A 6 m by 8 m rectangular garden has a diagonal path. How much shorter is that path than walking along one 6 m side and one 8 m side?
3 marks

Read the worked solutions

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

A rectangular screen is 12 cm by 16 cm. Find its diagonal.
2 marks
$d^{2} = 144 + 256 = 400$ .
Take the positive root.
20 cm.
A 17 m ladder has its foot 8 m from a perpendicular wall. Find the height reached.
2 marks
$h^{2} = 289 - 64 = 225$ .
The positive root is 15.
15 m.
A cable runs from the top of a 12 m pole to a point 5 m horizontally from its base. Find the taut cable length.
2 marks
Model a right triangle with legs 12 and 5.
$c^{2} = 144 + 25 = 169$ .
13 m.
A 25 m wire reaches a pole top 24 m above level ground. Find the horizontal distance of its ground anchor from the pole.
2 marks
$x^{2} = 625 - 576 = 49$ .
Take the root.
7 m.
A 15 m ladder moves from a foot distance of 9 m to a foot distance of 12 m from a wall. Find the vertical drop of its top.
3 marks
Initial height: $\sqrt{225 - 81} = 12$ m.
Final height: $\sqrt{225 - 144} = 9$ m; subtract.
3 m.
A 6 m by 8 m rectangular garden has a diagonal path. How much shorter is that path than walking along one 6 m side and one 8 m side?
3 marks
The diagonal is $\sqrt{36 + 64} = 10$ m.
The two sides total 14 m; subtract 10.
4 m shorter.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Hypotenuse and right-angled triangles; why the theorem works; finding the hypotenuse; finding a shorter side; exact roots versus rounded answers; checking whether a triangle is right-angled; practical length problems.

Official curriculum source ↗

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