Practical length problems | Year 8

01 / LEARN

Understand the idea.

Draw the model and identify the requested quantity

Begin a practical problem by identifying a right triangle in the situation. A perpendicular wall and level ground provide the right angle; a straight ladder is the hypotenuse. A rectangle's diagonal similarly cuts it into two right triangles.

Label known measurements and the requested length before calculating. Use a sum of squares for an unknown hypotenuse, or subtract a known leg square from the hypotenuse square for an unknown leg. Keep measurement units consistent.

Interpret the mathematical result

Interpret the result in the original situation. A ladder's final height differs from the distance its top has dropped: a drop requires both initial and final heights, followed by subtraction in the correct order.

Check the model's assumptions, including a perpendicular wall, level ground and a straight ladder of unchanged length. Give units and suitable accuracy, and compare the result with the original measurements to catch an implausible length.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A rectangular diagonal

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

Worked solution

1
Identify the diagonal as the hypotenuse.
$d^{2} = 30^{2} + 40^{2}$
Adjacent sides of a rectangle meet at a right angle, and the diagonal joins their opposite ends.
2
Calculate the squared diagonal.
$d^{2} = 900 + 1600 = 2500$
The squares of the perpendicular field dimensions add.
3
Take the positive square root.
$d = \sqrt{2500} = 50 m$
This gives the straight distance between opposite corners.
4
Check against the field dimensions.
$40 < 50 < 30 + 40$
A diagonal exceeds either side but is shorter than travelling along two adjacent sides.

Answer

50 m.

EXAMPLE 02

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?

Worked solution

1
Calculate the initial vertical height.
$h_{i} = \sqrt{10^{2} - 6^{2}} = \sqrt{64} = 8 m$
The 10 m ladder is the hypotenuse; subtract the initial horizontal distance square.
2
Calculate the final vertical height.
$h_{f} = \sqrt{10^{2} - 8^{2}} = \sqrt{36} = 6 m$
The ladder length stays fixed while its foot moves farther from the wall.
3
Subtract to find the downward movement.
$h_{i} - h_{f} = 8 - 6 = 2 m$
The question asks for the loss of height, rather than either height on its own.

Answer

2 m.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

12 m.

The 13 m ladder is the hypotenuse, so $h^{2} = 13^{2} - 5^{2} = 169 - 25 = 144$ . The positive root gives a height of $12 m$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Draw the model and identify the requested quantity

Interpret the mathematical result

See the method in action.

A rectangular diagonal

Identify the diagonal as the hypotenuse.

Calculate the squared diagonal.

Take the positive square root.

Check against the field dimensions.

A drop is a difference

Calculate the initial vertical height.

Calculate the final vertical height.

Subtract to find the downward movement.

Try it yourself.

Check your understanding

Keep practising