Multi-step right-triangle problems | Year 10

01 / LEARN

The idea, step by step

Choose one triangle at a time

Pythagoras links the three sides of one right-angled triangle. The hypotenuse is opposite the right angle, not automatically the side drawn at the top. Sketch the situation and mark a right angle before choosing a formula.

For two triangles, name the shared length. Find it from the triangle with enough information, then carry its unrounded value into the next calculation.

c^{2} = a^{2} + b^{2}

Finish the whole problem

After finding a side, read the question again: you may still need a perimeter, a drop or the total amount of material. A diagonal, a vertical height and a total distance describe different quantities.

Retain exact roots until the last step. Include the unit and check that a hypotenuse exceeds either leg.

A common trap

Adding box dimensions to get its diagonal.

Instead: Adding edges measures a different route. Use perpendicular squared lengths for a diagonal.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two linked triangles

A box is 12 cm long, 9 cm wide and 20 cm high. Find its space diagonal.

The base diagonal is $d = \sqrt{12^{2} + 9^{2}} = 15$ cm.
The base diagonal and height form a second right triangle.
The space diagonal is $\sqrt{15^{2} + 20^{2}} = 25$ cm.

25 cm

Example 2

Add the allowance

A cable joins the top of an 8 m pole to a ground point 6 m from its base. Allow 1 m extra for attaching it. How much cable is needed?

The taut length is $\sqrt{8^{2} + 6^{2}} = 10$ m.
The attachment allowance is additional cable, not a side of the triangle.
Add 1 m to 10 m.

11 m

Check your understanding

A rectangle is 5 m by 12 m. Find its diagonal.

Show answer

13 m

$d = \sqrt{5^{2} + 12^{2}} = 13$ m.

03 / WORKSHEET

Your turn

6 questions · 16 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the diagonal of a 9 cm by 12 cm rectangle.
2 marks
A cuboid has perpendicular edges 3 m, 4 m and 12 m. Find its space diagonal.
3 marks
A 13 m ladder has its foot 5 m from a wall. How high does it reach?
2 marks
A rectangular garden has diagonal 26 m and width 10 m. Find its perimeter.
3 marks
A symmetrical roof has span 16 m and rise 6 m. Find the total length of its two sloping edges.
3 marks
A student claims a right triangle with hypotenuse 17 cm and leg 8 cm has other leg 19 cm. Correct the result and explain.
3 marks

Read the worked solutions

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

Find the diagonal of a 9 cm by 12 cm rectangle.
2 marks
$d^{2} = 9^{2} + 12^{2} = 225$ .
$d = 15$ cm.
15 cm
A cuboid has perpendicular edges 3 m, 4 m and 12 m. Find its space diagonal.
3 marks
The base diagonal is $\sqrt{3^{2} + 4^{2}} = 5$ m.
The space diagonal is $\sqrt{5^{2} + 12^{2}} = 13$ m.
13 m
A 13 m ladder has its foot 5 m from a wall. How high does it reach?
2 marks
$h^{2} = 13^{2} - 5^{2} = 144$ .
$h = 12$ m.
12 m
A rectangular garden has diagonal 26 m and width 10 m. Find its perimeter.
3 marks
The other side is $\sqrt{26^{2} - 10^{2}} = 24$ m.
The perimeter is $2 (24 + 10) = 68$ m.
68 m
A symmetrical roof has span 16 m and rise 6 m. Find the total length of its two sloping edges.
3 marks
Each half has run 8 m.
Each slope is $\sqrt{8^{2} + 6^{2}} = 10$ m.
There are two slopes.
20 m
A student claims a right triangle with hypotenuse 17 cm and leg 8 cm has other leg 19 cm. Correct the result and explain.
3 marks
$x^{2} = 17^{2} - 8^{2} = 225$ .
$x = 15$ cm.
Subtract the known leg square from the hypotenuse square.
15 cm; a leg cannot exceed the hypotenuse.

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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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