OpenRevise

Exam questions · Maths · Geometry and Measures

Pythagoras' Theorem

  • 6 exam questions
  • 20 marks
  • 9 quick checks
  1. 1 Work out [4 marks]

    The diagram shows a rectangle. (a) Work out the length of the diagonal, \(x\). [3 marks] (b) Work out the area of the rectangle. [1 mark]

    A rectangle 15 cm by 8 cm with its diagonal marked x.
    Show answerHide answer

    Model answer

    (a) \(x^2 = 15^2 + 8^2 = 225 + 64 = 289\), so \(x = 17\) cm. (b) \(15 \times 8 = 120\) cm\(^2\).

    Mark scheme

    • (a) \(15^2 + 8^2\) — M1
    • (a) \(289\) — M1
    • (a) 17 cm — A1
    • (b) 120 cm\(^2\) — B1
  2. 2 Work out [3 marks]

    A right-angled triangle has shorter sides of length 10 cm and 24 cm. Work out the length of the hypotenuse.

    Show answerHide answer

    Model answer

    \(10^2 + 24^2 = 100 + 576 = 676\), and \(\sqrt{676} = 26\) cm.

    Mark scheme

    • \(10^2 + 24^2\) — M1
    • 676 — M1
    • 26 cm — A1
  3. 3 Work out [3 marks]

    A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Work out the length of the other shorter side.

    Show answerHide answer

    Model answer

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

    Mark scheme

    • \(17^2 - 8^2\) — M1
    • 225 — M1
    • 15 cm — A1
  4. 4 Work out [3 marks]

    A boat sails 12 km north and then 5 km east. Work out the distance of the boat from its starting point.

    Show answerHide answer

    Model answer

    The path makes a right-angled triangle, so the distance is \(\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\) km.

    Mark scheme

    • \(12^2 + 5^2\) — M1
    • 169 — M1
    • 13 km — A1
  5. 5 Work out [3 marks]

    A triangle has sides of length 10 cm, 24 cm and 26 cm. Is the triangle right-angled? You must show your working.

    Show answerHide answer

    Model answer

    \(10^2 + 24^2 = 100 + 576 = 676\) and \(26^2 = 676\). The squares match, so the triangle is right-angled.

    Mark scheme

    • \(10^2 + 24^2 = 676\) — M1
    • \(26^2 = 676\) — M1
    • Yes, with the two values compared — Q1
  6. 6 Work out [4 marks]

    A cuboid measures 3 cm by 4 cm by 12 cm. Work out the length of the longest straight line that can be drawn inside the cuboid, from one corner to the opposite corner.

    Show answerHide answer

    Model answer

    The diagonal of the 3 by 4 face is \(\sqrt{9 + 16} = 5\) cm. Then the space diagonal is \(\sqrt{5^2 + 12^2} = \sqrt{169} = 13\) cm.

    Mark scheme

    • \(3^2 + 4^2 = 25\) or 5 found — M1
    • \(5^2 + 12^2\) or \(3^2 + 4^2 + 12^2\) — M1
    • 169 — A1
    • 13 cm — 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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    A: 10

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