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Exam questions · Maths · Geometry and Measures

Pythagoras' Theorem

  • 7 exam questions
  • 23 marks
  • 9 quick checks
  1. 1 Calculate [4 marks]

    The diagram shows a right-angled triangle \(ABC\). (a) Calculate the length of \(AC\). [3 marks] (b) Calculate the perimeter of the triangle. [1 mark]

    A right-angled triangle ABC with the right angle at B, BC = 12 cm and AB = 9 cm.
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    Model answer

    (a) \(AC^2 = 9^2 + 12^2 = 81 + 144 = 225\), so \(AC = 15\) cm. (b) The perimeter is \(9 + 12 + 15 = 36\) cm.

    Mark scheme

    • (a) \(9^2 + 12^2\) — M1
    • (a) 225 — M1
    • (a) 15 cm — A1
    • (b) 36 cm — B1
  2. 2 Calculate [3 marks]

    A right-angled triangle has shorter sides of length 8 cm and 15 cm. Calculate the length of the hypotenuse.

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    Model answer

    \(8^2 + 15^2 = 64 + 225 = 289\), and \(\sqrt{289} = 17\) cm.

    Mark scheme

    • \(8^2 + 15^2\) — M1
    • 289 — M1
    • 17 cm — A1
  3. 3 Calculate [3 marks]

    A right-angled triangle has a hypotenuse of 41 cm and one shorter side of 9 cm. Calculate the length of the other shorter side.

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    Model answer

    \(41^2 - 9^2 = 1681 - 81 = 1600\), and \(\sqrt{1600} = 40\) cm.

    Mark scheme

    • \(41^2 - 9^2\) — M1
    • 1600 — M1
    • 40 cm — A1
  4. 4 Calculate [3 marks]

    A vertical flagpole stands on level ground. A wire 13 m long joins the top of the flagpole to a point on the ground 5 m from the base of the pole. Calculate the height of the flagpole.

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    Model answer

    \(h^2 = 13^2 - 5^2 = 169 - 25 = 144\), so \(h = 12\) m.

    Mark scheme

    • \(13^2 - 5^2\) — M1
    • 144 — M1
    • 12 m — A1
  5. 5 Calculate [3 marks]

    A rhombus has diagonals of length 16 cm and 12 cm. Calculate the length of one side of the rhombus.

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    Model answer

    The diagonals of a rhombus cross at right angles and bisect each other, so a right-angled triangle has shorter sides of 8 cm and 6 cm. The side is \(\sqrt{8^2 + 6^2} = \sqrt{100} = 10\) cm.

    Mark scheme

    • Half-diagonals 8 and 6 used — M1
    • \(8^2 + 6^2 = 100\) — M1
    • 10 cm — A1
  6. 6 Calculate [4 marks]

    An isosceles triangle has a base of 12 cm and two equal sides of 10 cm. Calculate the area of the triangle.

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    Model answer

    The height splits the base into two lots of 6 cm, so \(h = \sqrt{10^2 - 6^2} = \sqrt{64} = 8\) cm. The area is \(\dfrac{1}{2} \times 12 \times 8 = 48\) cm\(^2\).

    Mark scheme

    • Half of the base \(= 6\) used — M1
    • \(10^2 - 6^2 = 64\) — M1
    • \(h = 8\) — A1
    • 48 cm\(^2\) — A1
  7. 7 Calculate [3 marks]

    \(P\) is the point \((2, -1)\) and \(Q\) is the point \((9, 23)\). Calculate the length of \(PQ\).

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    Model answer

    The horizontal difference is 7 and the vertical difference is \(23 - (-1) = 24\). So \(PQ = \sqrt{7^2 + 24^2} = \sqrt{625} = 25\).

    Mark scheme

    • Differences 7 and 24 found — M1
    • \(7^2 + 24^2 = 625\) — M1
    • 25 — A1

Quick check

  1. 1

    A right-angled triangle has shorter sides of 9 cm and 12 cm. What is the hypotenuse?

    1. A15 cm
    2. B21 cm
    3. C225 cm
    4. D10 cm
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    A: 15 cm

    \(9^2 + 12^2 = 81 + 144 = 225\), and \(\sqrt{225} = 15\).

  2. 2

    A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. What is the other shorter side?

    1. A8 cm
    2. B18 cm
    3. C144 cm
    4. D12 cm
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    D: 12 cm

    \(13^2 - 5^2 = 169 - 25 = 144\), and \(\sqrt{144} = 12\).

  3. 3

    Which set of lengths makes a right-angled triangle?

    1. A5 cm, 7 cm, 9 cm
    2. B6 cm, 7 cm, 10 cm
    3. C6 cm, 8 cm, 10 cm
    4. D4 cm, 5 cm, 7 cm
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    C: 6 cm, 8 cm, 10 cm

    \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\). The other sets do not satisfy \(a^2 + b^2 = c^2\).

  4. 4

    Which side of a right-angled triangle is the hypotenuse?

    1. AThe shortest side
    2. BThe longest side, opposite the right angle
    3. CThe side along the bottom
    4. DThe side next to the right angle
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    B: The longest side, opposite the right angle

    The hypotenuse is always the longest side, and it is opposite the right angle.

  5. 5

    A rectangle is 15 cm long and 8 cm wide. How long is its diagonal?

    1. A17 cm
    2. B23 cm
    3. C7 cm
    4. D289 cm
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    A: 17 cm

    \(15^2 + 8^2 = 225 + 64 = 289\), and \(\sqrt{289} = 17\).

  6. 6

    A ladder 6.5 m long leans against a wall. Its foot is 2.5 m from the wall. How high up the wall does it reach?

    1. A9 m
    2. B4 m
    3. C36 m
    4. D6 m
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    D: 6 m

    \(6.5^2 - 2.5^2 = 42.25 - 6.25 = 36\), and \(\sqrt{36} = 6\).

  7. 7

    An isosceles triangle has base 10 cm and equal sides of 13 cm. What is its height?

    1. A8 cm
    2. B9 cm
    3. C12 cm
    4. D7 cm
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    C: 12 cm

    The height splits the base into two lots of 5 cm. \(13^2 - 5^2 = 144\), so the height is 12 cm.

  8. 8

    A right-angled triangle has shorter sides of 2 cm and 4 cm. What is the hypotenuse?

    1. A\(2\sqrt{3}\) cm
    2. B\(2\sqrt{5}\) cm
    3. C6 cm
    4. D\(\sqrt{6}\) cm
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    B: \(2\sqrt{5}\) cm

    \(2^2 + 4^2 = 20\), and \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\).

  9. 9

    What is the distance between the points \((1, 2)\) and \((7, 10)\)?

    1. A10
    2. B14
    3. C100
    4. D8
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    A: 10

    The horizontal difference is 6 and the vertical difference is 8, so the distance is \(\sqrt{6^2 + 8^2} = \sqrt{100} = 10\).