Robinson Tuition
Grade 5 · Foundation and Higher · free GCSE maths worksheet

Similar shapes: finding lengths

Using scale factors.

14 original questions, 47 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Rectangle A is 4 cm by 6 cm. Rectangle B is 8 cm by 10 cm.

Sam says, "Each side of rectangle B is 4 cm longer than the matching side of rectangle A, so the rectangles are similar."

Explain why Sam is wrong.

1 mark
Show answer
Similar shapes need the same scale factor (ratio) for every pair of sides, not the same added amount: 8 divided by 4 = 2 but 10 divided by 6 is not 2, so the rectangles are not similar
2

Triangle ABC is similar to triangle PQR.

AB corresponds to PQ, and BC corresponds to QR.

AB = 6 cm, PQ = 9 cm and BC = 8 cm.

Work out the length of QR.

2 marks
Show answer
12 cm
3

Triangle ABC is similar to triangle PQR.

AB corresponds to PQ, BC corresponds to QR and AC corresponds to PR.

AB : PQ = 3 : 7

AC = 15 cm and BC = 9 cm.

  1. Work out the length of PR.
  2. Work out the length of QR.
3 marks
Show answer
a) 35 cm b) 21 cm
4

In triangle ABC, angle A = 35° and angle B = 80°.

In triangle DEF, angle D = 80° and angle E = 65°.

Show that triangle ABC is similar to triangle DEF.

2 marks
Show answer
Angle C = 180 - 35 - 80 = 65 degrees, so both triangles have angles 35, 80 and 65 degrees; triangles with the same three angles are similar
5

A photograph is 15 cm long and 10 cm wide.

The photograph is enlarged to make a print that is mathematically similar to it.

The print is 24 cm long.

  1. Work out the scale factor of the enlargement.
  2. Work out the width of the print.
3 marks
Show answer
a) 1.6 b) 16 cm
6

Triangle ABC is isosceles with AB = AC = 9 cm and BC = 6 cm.

Triangle DEF is isosceles and is similar to triangle ABC, with DE = DF = 15 cm.

  1. Work out the scale factor from triangle ABC to triangle DEF.
  2. Work out the length of EF.
3 marks
Show answer
a) 5/3 b) 10 cm
7

Triangle ABC is similar to triangle DEF.

AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF.

AB = 5 cm, BC = 7 cm, DE = 7.5 cm and DF = 12 cm.

  1. Work out the length of EF.
  2. Work out the length of AC.
4 marks
Show answer
a) 10.5 cm b) 8 cm
8

Triangle ABC has sides AB = 4 cm, BC = 6 cm and AC = 8 cm.

Triangle DEF has sides DE = 6 cm, EF = 9 cm and DF = 12 cm.

  1. Show that triangle ABC is similar to triangle DEF.
  2. Triangle PQR is similar to triangle ABC. PQ corresponds to AB and PQ = 10 cm. Work out the perimeter of triangle PQR.
  3. Triangle GHI is congruent to triangle ABC. Write down the scale factor from triangle ABC to triangle GHI.
5 marks
Show answer
a) all three pairs of sides are in the ratio 1.5 b) 45 cm c) 1
9

In triangle ABC, D is a point on AB and E is a point on AC.

DE is parallel to BC.

AD = 4 cm, DB = 6 cm and DE = 5 cm.

Work out the length of BC.

4 cm6 cm5 cmABCDEDiagram NOT accurately drawn
3 marks
Show answer
12.5 cm
10

AB is parallel to DE.

The straight lines AE and BD cross at the point C.

AC = 9 cm, CE = 6 cm, AB = 12 cm and CD = 5 cm.

  1. Work out the length of DE.
  2. Work out the length of BC.
12 cm9 cm6 cm5 cmABCDEDiagram NOT accurately drawn
4 marks
Show answer
a) 8 cm b) 7.5 cm
11

Triangle ABC is similar to triangle DEF.

AB corresponds to DE and BC corresponds to EF.

AB = 6 cm, DE = 9 cm, BC = x cm and EF = (x + 4) cm.

Work out the value of x and the length of EF.

4 marks
Show answer
a) x = 8 b) EF = 12 cm
12

In triangle ADE, B is a point on AD and C is a point on AE.

BC is parallel to DE.

AB = 6 cm, BD = 3 cm and BC = 8 cm.

The area of triangle ABC is 24 cm2.

  1. Work out the length of DE.
  2. Work out the area of the trapezium BDEC.
6 cm3 cm8 cmADEBCDiagram NOT accurately drawn
4 marks
Show answer
a) 12 cm b) 30 cm²
13

In triangle PQR, angle PQR = 90°, PQ = 6 cm, QR = 8 cm and PR = 10 cm.

S is the point on PR such that QS is perpendicular to PR.

Work out the length of QS.

6 cm8 cm10 cmPQRSDiagram NOT accurately drawn
4 marks
Show answer
4.8 cm
14

In triangle ABC, D is a point on AB.

Angle ACD = angle ABC.

AC = 8 cm and AD = 4 cm.

  1. Prove that triangle ACD is similar to triangle ABC.
  2. Work out the length of BD.
4 cm8 cmABCDDiagram NOT accurately drawn
5 marks
Show answer
a) shared angle A and angle ACD = angle ABC, so the triangles have two equal angles and are similar b) 12 cm

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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