[discia] GCSE Maths › Geometry & Measures › Pythagoras

Pythagoras

discia.org 10 questions · 31 marks
Name Date
  1. Calculator 3 marks

    Here is a right-angled triangle.

    15 cmx cm8 cmDiagram NOT accurately drawn

    Calculate the length of the side marked \(x\).

    Show answer
    \(17\ \text{cm}\)
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  2. Calculator 3 marks

    Here is a right-angled triangle.

    24 cm30 cmx cmDiagram NOT accurately drawn

    Work out the length of the side marked \(x\).

    Show answer
    \(18\ \text{cm}\)
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  3. Calculator 3 marks

    \(ABCD\) is a rectangle.

    16 cm30 cmABCDDiagram NOT accurately drawn

    Work out the length of the diagonal \(AC\).

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    \(34\ \text{cm}\)
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  4. Calculator 5 marks

    \(ABCD\) is a rectangle. The dashed line is the diagonal \(AC\).

    24 cm45 cmABCDDiagram NOT accurately drawn
    (a)

    Calculate the length of \(AC\).

    (b)

    Calculate the area of triangle \(ABC\).

    Show answer
    (a) \(51\ \text{cm}\)
    (b) \(540\ \text{cm}^2\)
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  5. Calculator 3 marks

    The diagram shows a right-angled triangle.

    0.3 cm0.5 cmx cmDiagram NOT accurately drawn

    Use Pythagoras' theorem to show that \(x = 0.4\)

    Show answer
    \(x^2 = 0.25 - 0.09 = 0.16 = 0.4^2\), so \(x = 0.4\) cm
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  6. Calculator 2 marks

    Here is a right-angled triangle.

    36 cm3y cm15 cm

    Kai works out that \(y = 13\).

    Mel says, "If \(y = 13\), the hypotenuse would be shorter than both of the other sides, so \(y\) cannot be \(13\)."

    Is Mel correct?

    Tick a box. Yes / No

    Give a reason for your answer.

    Show answer
    No. With \(y = 13\) the label \(3y\) cm gives a hypotenuse of \(39\) cm, which is longer than the other sides, \(15\) cm and \(36\) cm.
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  7. Calculator 1 mark

    The diagram shows a right-angled triangle.

    7 cm2y cm24 cmRPQ

    Noah is using Pythagoras' theorem to find the value of \(y\). Here is his working.

    \(PQ^2 = RP^2 + QR^2\)

    \(2y^2 = 7^2 + 24^2\)

    \(2y^2 = 49 + 576 = 625\)

    \(y = \sqrt{\frac{625}{2}}\)

    Here are four statements about Noah's working.

    1. Noah has squared only part of the length of the hypotenuse. 2. Noah has added the two shorter sides instead of square rooting the total of their squares. 3. Noah has treated one of the shorter sides as the hypotenuse. 4. Noah has made a slip when adding two numbers.

    What mistake has Noah made? Write down the number of the correct statement.

    Show answer
    Statement 1: Noah has squared only part of the length of the hypotenuse.
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  8. Calculator 3 marks

    A triangle has sides of length 0.6 cm, 2.5 cm and 2.4 cm.

    Is the triangle right-angled? You must show your working.

    Show answer
    No: \(0.6^2 + 2.4^2 = 6.12\) and \(2.5^2 = 6.25\), and 6.12 is less than 6.25, so the angle opposite the longest side is more than 90° and the triangle is not right-angled
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  9. Calculator 5 marks

    \(ABC\) is an isosceles triangle with \(AB = AC\).

    40 cm29 cmBCA
    (a)

    Work out the perpendicular height of the triangle from \(A\) to \(BC\).

    (b)

    Work out the area of triangle \(ABC\).

    Show answer
    (a) \(21\ \text{cm}\)
    (b) \(420\ \text{cm}^2\)
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  10. Calculator 3 marks

    Amara sails her yacht 8 km due west, then turns and sails 6 km due north.

    Calculate how far she is from the starting point.

    Show answer
    \(10\ \text{km}\)
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discia.org Pythagoras · sheet 1
Answers Pythagoras · sheet 1
  1. \(17\ \text{cm}\)
  2. \(18\ \text{cm}\)
  3. \(34\ \text{cm}\)
  4. (a) \(51\ \text{cm}\)
    (b) \(540\ \text{cm}^2\)
  5. \(x^2 = 0.25 - 0.09 = 0.16 = 0.4^2\), so \(x = 0.4\) cm
  6. No. With \(y = 13\) the label \(3y\) cm gives a hypotenuse of \(39\) cm, which is longer than the other sides, \(15\) cm and \(36\) cm.
  7. Statement 1: Noah has squared only part of the length of the hypotenuse.
  8. No: \(0.6^2 + 2.4^2 = 6.12\) and \(2.5^2 = 6.25\), and 6.12 is less than 6.25, so the angle opposite the longest side is more than 90° and the triangle is not right-angled
  9. (a) \(21\ \text{cm}\)
    (b) \(420\ \text{cm}^2\)
  10. \(10\ \text{km}\)
discia.org Pythagoras · sheet 1