Exact/rounded missing lengths | Year 9

01 / LEARN

The idea, step by step

Exact means no information is lost

An exact answer such as √74 names a length without rounding. Its calculator display is approximate if the root is irrational. You may keep a square-root form without simplifying the surd.

Use the positive root because a length is positive. Keep the exact expression or full calculator value if another calculation follows.

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

Choose the requested precision

For two decimal places, keep the hundredths digit and use the next digit to decide whether to round up. Use approximately when replacing an exact expression by a rounded decimal.

Extra calculator digits do not make an approximate measurement exact. Nevertheless, repeatedly rounding intermediate answers can make the final answer less accurate.

A common trap

Rounding each side or intermediate root before the final calculation.

Instead: Keep the original measurements and full intermediate values; round the requested final length.

02 / WORKED EXAMPLES

See the thinking

Example 1

An irrational hypotenuse

Find the hypotenuse with legs 5 cm and 7 cm, exactly and to two decimal places.

$c^{2} = 25 + 49 = 74$ .
$c = \sqrt{74} \approx 8.602325 \dots$ .

√74 cm; approximately 8.60 cm

Example 2

Subtract first

A right triangle has hypotenuse 13 m and one leg 9 m. Find the other leg exactly and to two decimal places.

$x^{2} = 169 - 81 = 88$ .
$x = \sqrt{88} \approx 9.380831 \dots$ .

√88 m; approximately 9.38 m

Check your understanding

Find the exact hypotenuse with legs 2 and 3.

Show answer

$\sqrt{13}$

$c^{2} = 4 + 9 = 13$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find the hypotenuse with legs 4 cm and 6 cm, exactly and to two decimal places.
3 marks
Legs are 9 m and 12 m. Is 15 m an exact hypotenuse? Explain.
2 marks
Find the missing leg with hypotenuse 10 cm and other leg 6 cm.
2 marks
Find the missing leg with hypotenuse 15 m and other leg 10 m, exactly and to two decimal places.
3 marks
Legs are 4.3 m and 6.8 m. Find the hypotenuse to two decimal places.
3 marks
A 2 m by 3 m rectangle has diagonal √13 m. Why is 3.61 m only approximate?
2 marks

Read the worked solutions

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

Find the hypotenuse with legs 4 cm and 6 cm, exactly and to two decimal places.
3 marks
$c = \sqrt{16 + 36} = \sqrt{52} \approx 7.2111 \dots$ cm.
$\sqrt{52}$ cm; 7.21 cm
Legs are 9 m and 12 m. Is 15 m an exact hypotenuse? Explain.
2 marks
$c = \sqrt{81 + 144} = \sqrt{225} = 15$ m.
The root is an integer, so no rounding occurred.
Yes
Find the missing leg with hypotenuse 10 cm and other leg 6 cm.
2 marks
$x = \sqrt{100 - 36} = \sqrt{64} = 8$ cm.
8 cm
Find the missing leg with hypotenuse 15 m and other leg 10 m, exactly and to two decimal places.
3 marks
$x = \sqrt{225 - 100} = \sqrt{125} \approx 11.1803 \dots$ m.
$\sqrt{125}$ m; 11.18 m
Legs are 4.3 m and 6.8 m. Find the hypotenuse to two decimal places.
3 marks
$c^{2} = 18.49 + 46.24 = 64.73$ .
$c = \sqrt{64.73} \approx 8.0455 \dots$ m.
8.05 m
A 2 m by 3 m rectangle has diagonal √13 m. Why is 3.61 m only approximate?
2 marks
$d^{2} = 2^{2} + 3^{2} = 13$ .
3.61 is rounded; √13 retains the exact value.
√13 is irrational, so its decimal does not terminate.

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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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