Pythagoras
Geometry and measures
- Crossover
- Non-calculator
- 8 questions
- 29 marks
Questions marked Higher are aimed at the Higher tier.
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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A right-angled triangle has legs of $3$ cm and $4$ cm. Find the hypotenuse.
Right-angled triangle with legs 3 cm and 4 cm 2 marks
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A right-angled triangle has legs of $5$ cm and $12$ cm. Find the hypotenuse.
2 marks
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The hypotenuse is $15$ cm and one leg is $9$ cm. Find the other leg.
3 marks
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Which of these triangles is right-angled: sides $7$, $8$, $11$, or sides $6$, $8$, $10$?
3 marks
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A rectangle is $9$ cm by $12$ cm. Find the length of a diagonal.
4 marks
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A ladder is $6.5$ m long. Its foot is $2.5$ m from a vertical wall. How high up the wall does the ladder reach?
4 marks
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Find the distance from $(1, 2)$ to $(6, 14)$. An isosceles triangle has equal sides of $13$ cm and base $10$ cm. Find its height and area. Higher
5 marks
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A rectangle is $20$ m by $21$ m. Find its diagonal. A $12$ m pole has a rope from the top to a point $9$ m from the base, then the rope is moved to a point $5$ m from the base. Find both rope lengths and how much shorter the second rope is. Higher
6 marks
Mark scheme
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Answer. $5$ cm
- $\sqrt{3^{2} + 4^{2}} = \sqrt{25}$
- $5$ cm
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Answer. $13$ cm
- $\sqrt{25 + 144} = \sqrt{169}$
- $13$ cm
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Answer. $12$ cm
- $\sqrt{15^{2} - 9^{2}} = \sqrt{144}$
- $12$ cm
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Answer. Only $6$, $8$, $10$
- $7^{2} + 8^{2} = 113$, which is not $11^{2}$.
- $6^{2} + 8^{2} = 10^{2}$, so this triangle is right-angled.
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Answer. $15$ cm
- $\sqrt{9^{2} + 12^{2}} = \sqrt{225}$
- $15$ cm
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Answer. $6$ m
- $\sqrt{6.5^{2} - 2.5^{2}} = \sqrt{36}$
- $6$ m
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Answer. Distance $13$. Height $12$ cm and area $60$ cm$^{2}$.
- The coordinate differences are $5$ and $12$, so the distance is $13$.
- Height $= \sqrt{13^{2} - 5^{2}} = 12$ cm.
- Area $= \dfrac{1}{2} \times 10 \times 12 = 60$ cm$^{2}$.
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Answer. Diagonal $29$ m. Rope lengths $15$ m and $13$ m, so the second is $2$ m shorter.
- $\sqrt{20^{2} + 21^{2}} = \sqrt{841} = 29$ m.
- $\sqrt{12^{2} + 9^{2}} = 15$ m.
- $\sqrt{12^{2} + 5^{2}} = 13$ m.
- The difference is $2$ m.