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Similar Shapes (Lengths)

discia.org 10 questions · 31 marks
Name Date
  1. Either paper 1 mark

    Here are five rectangles.

    21 cm12 cmXDiagram NOT accurately drawn 23 cm14 cmADiagram NOT accurately drawn 14 cm9 cmBDiagram NOT accurately drawn 21 cm10 cmCDiagram NOT accurately drawn 28 cm16 cmDDiagram NOT accurately drawn

    Which shape is similar to shape \(X\)?

    Circle the correct letter.

    Show answer
    D
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  2. Either paper 2 marks

    Here are two rectangles, \(A\) and \(B\). Each is drawn to fill its own frame, so the sizes on the page are not to the same scale.

    15 cm9 cmA 3 cm5 cmB

    Show that these two rectangles are similar.

    Show answer
    Long sides \(5 : 15 = 1 : 3\); short sides \(3 : 9 = 1 : 3\). The ratios are equal, so the rectangles are similar.
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  3. Either paper 4 marks

    Rectangle \(A\) and rectangle \(B\) are similar.

    11 cm4 cmADiagram NOT accurately drawn 44 cmx cmBDiagram NOT accurately drawn
    (a)

    Work out the scale factor of the enlargement from rectangle \(A\) to rectangle \(B\).

    (b)

    Work out the length marked \(x\).

    Show answer
    (a) \(4\)
    (b) \(16\ \text{cm}\)
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  4. Either paper 3 marks

    Triangle \(ABC\) is similar to triangle \(PQR\), so that \(A\) corresponds to \(P\), \(B\) to \(Q\) and \(C\) to \(R\).

    7 cm8 cmABCDiagram NOT accurately drawn 14 cmx cmPQRDiagram NOT accurately drawn

    Find the length of \(QR\).

    Show answer
    \(16\ \text{cm}\)
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  5. Either paper 3 marks

    Priya has a photograph that is \(6\) cm wide and \(9\) cm high.

    She enlarges it. The enlarged photograph is \(14\) cm wide, and it is similar to the original.

    Calculate the height of the enlarged photograph.

    Show answer
    \(21\ \text{cm}\)
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  6. Either paper 3 marks

    One afternoon Priya measures the shadows in a park.

    A bollard \(2\) m tall casts a shadow \(5\) m long. At the same moment a tree casts a shadow \(25\) m long.

    The sun's rays are parallel, so the two triangles below are similar.

    5 m2 mDiagram NOT accurately drawn 25 mx mDiagram NOT accurately drawn

    Find the height of the tree.

    Show answer
    \(10\ \text{m}\)
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  7. Either paper 4 marks

    Triangle \(ABC\) is similar to triangle \(PQR\), so that \(A\) corresponds to \(P\), \(B\) to \(Q\) and \(C\) to \(R\).

    \(AB = 9\) cm and \(PQ = 36\) cm.

    Dev says:

    "The scale factor of the enlargement from \(ABC\) to \(PQR\) is \(27\), because \(36 - 9 = 27\)."

    (a)

    Explain why Dev is wrong.

    (b)

    State the correct scale factor.

    (c)

    \(BC = 10\) cm. Work out the length of \(QR\).

    Show answer
    (a) A scale factor multiplies rather than adds, so it comes from dividing corresponding lengths: \(36 \div 9 = 4\), not \(36 - 9 = 27\).
    (b) \(4\)
    (c) \(40\ \text{cm}\)
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  8. Either paper 6 marks

    Triangle \(ABC\) is similar to triangle \(PQR\), so that \(A\) corresponds to \(P\), \(B\) to \(Q\) and \(C\) to \(R\).

    27 cmx cm21 cmABCDiagram NOT accurately drawn 9 cm8 cmy cmPQRDiagram NOT accurately drawn
    (a)

    Write down, as a fraction in its simplest form, the scale factor of the enlargement from triangle \(ABC\) to triangle \(PQR\).

    (b)

    Work out the length marked \(y\).

    (c)

    Work out the length marked \(x\).

    Show answer
    (a) \(\frac{1}{3}\)
    (b) \(7\ \text{cm}\)
    (c) \(24\ \text{cm}\)
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  9. Either paper 3 marks

    The diagram shows two similar right-angled triangles, \(A\) and \(B\).

    12 cmh cmTriangle ADiagram NOT accurately drawn 20 cmTriangle BDiagram NOT accurately drawn

    The area of triangle \(B\) is 150 cm².

    Calculate the value of \(h\).

    Show answer
    \(9\ \text{cm}\)
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  10. Either paper 2 marks

    The diagram shows triangle \(FGH\) and triangle \(FJK\).

    \(J\) lies on \(FG\) and \(K\) lies on \(FH\). \(JK\) is parallel to \(GH\).

    FJGHK

    Show that triangle \(FGH\) and triangle \(FJK\) are similar.

    Give a reason for each step of your working.

    Show answer
    Angle \(GFH\) is the same as angle \(JFK\), because it is the angle at \(F\) shared by both triangles. Angle \(FGH\) = angle \(FJK\), because they are corresponding angles on the parallel lines \(JK\) and \(GH\). Angle \(FHG\) = angle \(FKJ\), because they are corresponding angles on the same parallel lines (or the third pair is equal by the angle sum). Two pairs of angles are equal, so the triangles are similar.
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discia.org Similar Shapes (Lengths) · sheet 1
Answers Similar Shapes (Lengths) · sheet 1
  1. D
  2. Long sides \(5 : 15 = 1 : 3\); short sides \(3 : 9 = 1 : 3\). The ratios are equal, so the rectangles are similar.
  3. (a) \(4\)
    (b) \(16\ \text{cm}\)
  4. \(16\ \text{cm}\)
  5. \(21\ \text{cm}\)
  6. \(10\ \text{m}\)
  7. (a) A scale factor multiplies rather than adds, so it comes from dividing corresponding lengths: \(36 \div 9 = 4\), not \(36 - 9 = 27\).
    (b) \(4\)
    (c) \(40\ \text{cm}\)
  8. (a) \(\frac{1}{3}\)
    (b) \(7\ \text{cm}\)
    (c) \(24\ \text{cm}\)
  9. \(9\ \text{cm}\)
  10. Angle \(GFH\) is the same as angle \(JFK\), because it is the angle at \(F\) shared by both triangles. Angle \(FGH\) = angle \(FJK\), because they are corresponding angles on the parallel lines \(JK\) and \(GH\). Angle \(FHG\) = angle \(FKJ\), because they are corresponding angles on the same parallel lines (or the third pair is equal by the angle sum). Two pairs of angles are equal, so the triangles are similar.
discia.org Similar Shapes (Lengths) · sheet 1