Year 9 Pythagoras' theorem and trigonometry: free practice test with answers

Sample pythagoras' theorem and trigonometry questions with answers

Try each one before you open the answer.

1. Pythagoras' theorem

   A right-angled triangle has shorter sides of $7$ cm and $24$ cm. Find the length of the hypotenuse.

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   Key idea: Use $c^2 = a^2 + b^2$.

   1. $c^2 = 7^2 + 24^2 = 49 + 576 = 625$
   2. $c = \sqrt{625} = 25$

   Answer: $25$ cm

2. Finding the length of the shorter sides

   A right-angled triangle has a hypotenuse of $17$ cm and one shorter side of $8$ cm. Find the length of the third side.

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   Key idea: To find a shorter side, subtract: $a^2 = c^2 - b^2$, where $c$ is the hypotenuse.

   1. $a^2 = 17^2 - 8^2 = 289 - 64 = 225$
   2. $a = \sqrt{225} = 15$

   Answer: $15$ cm

3. Using Pythagoras' theorem to solve two-dimensional problems

   A $4.5$ m ladder leans against a vertical wall on flat ground. The foot of the ladder is $1.2$ m from the wall. How far up the wall does the ladder reach? Give your answer correct to $2$ decimal places.

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   Key idea: The ladder, wall and ground form a right-angled triangle with the ladder as the hypotenuse.

   1. $h^2 = 4.5^2 - 1.2^2 = 20.25 - 1.44 = 18.81$
   2. $h = \sqrt{18.81} = 4.337\ldots \approx 4.34$

   Answer: $4.34$ m

4. Using Pythagoras' theorem to solve three-dimensional problems

   A rectangular box has length $12$ cm, width $5$ cm and height $8$ cm. Find the length of the longest diagonal of the box, correct to one decimal place.

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   Key idea: The longest diagonal of a rectangular prism goes from one corner to the opposite corner. Use Pythagoras twice: first on the base, then use that result with the height.

   1. Find the diagonal $d$ of the base using Pythagoras:\

   $$\begin{gathered}d^2 = 12^2 + 5^2 = 144 + 25 = 169 \\ d = 13 \text{ cm}\end{gathered}$$

   2. Use $d$ and the height to find the space diagonal $D$:\

   $$\begin{gathered}D^2 = d^2 + 8^2 = 169 + 64 = 233 \\ D = \sqrt{233} \approx 15.3 \text{ cm}\end{gathered}$$

   Answer: The longest diagonal is approximately $15.3$ cm.

5. Finding unknown side lengths

   A right-angled triangle has an acute angle of $35°$. The hypotenuse is $12$ cm. Find the side opposite the $35°$ angle, correct to two decimal places.

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   Key idea: Identify the sides relative to the given angle, then choose the ratio that links those two sides.

   1. The known side is the hypotenuse ($12$ cm). The unknown side is opposite the $35°$ angle. Use sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
   2. Substitute the known values: $\sin 35° = \frac{x}{12}$
   3. Solve for $x$: $x = 12 \times \sin 35°$
   4. Calculate: $x = 12 \times 0.5736 \approx 6.88 \text{ cm}$

   Answer: The opposite side is approximately $6.88$ cm.

6. Solving for the denominator

   In a right-angled triangle, the side opposite a $32^\circ$ angle is $8$ cm. Find the length of the hypotenuse, $x$ cm, correct to $2$ decimal places.

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   Key idea: When the unknown is in the denominator, write the ratio, multiply both sides by $x$, then divide by the trigonometric value.

   1. $\sin 32^\circ = \frac{8}{x}$
   2. $x \sin 32^\circ = 8$
   3. $x = \frac{8}{\sin 32^\circ} = 15.096\ldots \approx 15.10$

   Answer: $x = 15.10$

7. Finding unknown angles

   A loading ramp rises $1.4$ m vertically over a horizontal distance of $5$ m. Find the angle the ramp makes with the horizontal, correct to the nearest degree.

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   Key idea: The vertical rise is opposite the unknown angle and the horizontal distance is adjacent to it, so use inverse tangent.

   1. Write the tangent ratio: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.4}{5}$
   2. Simplify: $\tan\theta = 0.28$
   3. Apply inverse tangent: $\theta = \tan^{-1}(0.28)$
   4. Calculate: $\theta \approx 15.64° \approx 16°$

   Answer: The ramp makes an angle of approximately $16°$ with the horizontal.

8. Applying trigonometry

   A $6.5$ m ladder leans against a vertical wall. The base of the ladder is on horizontal ground and sits $2.5$ m from the base of the wall. A safety rule says the angle the ladder makes with the ground must be between $70°$ and $80°$. Use trigonometry to find the angle the ladder makes with the ground, correct to one decimal place, and state whether the ladder is positioned safely.

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   Key idea: The ladder, wall and ground form a right-angled triangle. Use the cosine ratio, since the adjacent side (ground distance) and hypotenuse (ladder) are known, to find the angle at the base.

   1. Identify the sides relative to the angle at the ground: adjacent $= 2.5$ m, hypotenuse $= 6.5$ m.
   2. Write the cosine ratio:\

   $$\cos(\theta) = \frac{2.5}{6.5}$$

   3. Solve for $\theta$:\

   $$\theta = \cos^{-1}\!\left(\frac{2.5}{6.5}\right) \approx \cos^{-1}(0.3846) \approx 67.4°$$

   4. Compare with the safe range of $70°$ to $80°$: $67.4° < 70°$, so the ladder is outside the safe range.

   Answer: The ladder makes an angle of approximately $67.4°$ with the ground, which is not safe.

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 9 pythagoras (applied) worksheet
• Year 9 trigonometry worksheet