Similarity Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Similarity questions. See exactly how to solve problems on similar triangles, scale factor, exact fraction, reduction.

similar trianglesscale factorexact fractionreductionreverse scale factormissing length
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
Triangle ABCABC is similar to triangle PQRPQR, with AA corresponding to PP, BB to QQ and CC to RR. AB=4AB = 4 cm and PQ=12PQ = 12 cm. Work out the scale factor of the enlargement from triangle ABCABC to triangle PQRPQR.

Worked solution

  1. Write the scale factor as a fraction

    k=PQAB=124k = \frac{PQ}{AB} = \frac{12}{4}

    Divide the new length by the old length — new over old, in that order.

  2. Simplify the fraction

    k=3k = 3

    Cancelling gives the exact scale factor 33. Leave it as a fraction rather than a rounded decimal.

  3. State the scale factor of the enlargement

    ABC→PQR:k=3ABC \rightarrow PQR: \quad k = 3

    So the scale factor from triangle ABCABC to triangle PQRPQR is 33.

Answer
k=3k = 3
Question 2
1 markeasy
Triangle ABCABC is similar to triangle PQRPQR, with AA corresponding to PP, BB to QQ and CC to RR. AB=6AB = 6 cm and PQ=15PQ = 15 cm. Work out the scale factor of the enlargement from triangle ABCABC to triangle PQRPQR.

Worked solution

  1. Write the scale factor as a fraction

    k=PQAB=156k = \frac{PQ}{AB} = \frac{15}{6}

    Divide the new length by the old length — new over old, in that order.

  2. Simplify the fraction

    k=52k = \frac{5}{2}

    Cancelling gives the exact scale factor 52\frac{5}{2}. Leave it as a fraction rather than a rounded decimal.

  3. State the scale factor of the enlargement

    ABC→PQR:k=52ABC \rightarrow PQR: \quad k = \frac{5}{2}

    So the scale factor from triangle ABCABC to triangle PQRPQR is 52\frac{5}{2}.

Answer
k=52k = \frac{5}{2}
Question 3
1 markeasy
Triangle ABCABC is similar to triangle PQRPQR, with AA corresponding to PP, BB to QQ and CC to RR. AB=10AB = 10 cm and PQ=4PQ = 4 cm. Work out the scale factor of the enlargement from triangle ABCABC to triangle PQRPQR.

Worked solution

  1. Write the scale factor as a fraction

    k=PQAB=410k = \frac{PQ}{AB} = \frac{4}{10}

    Divide the new length by the old length — new over old, in that order.

  2. Simplify the fraction

    k=25k = \frac{2}{5}

    Cancelling gives the exact scale factor 25\frac{2}{5}. Leave it as a fraction rather than a rounded decimal.

  3. State the scale factor of the enlargement

    ABC→PQR:k=25ABC \rightarrow PQR: \quad k = \frac{2}{5}

    So the scale factor from triangle ABCABC to triangle PQRPQR is 25\frac{2}{5}.

Answer
k=25k = \frac{2}{5}
Question 4
2 markseasy
Triangle ABCABC is similar to triangle PQRPQR, with AA corresponding to PP, BB to QQ and CC to RR. AB=8AB = 8 cm and PQ=20PQ = 20 cm. Work out the scale factor of the enlargement from triangle PQRPQR to triangle ABCABC.

Worked solution

  1. Write the scale factor as a fraction

    k=ABPQ=820k = \frac{AB}{PQ} = \frac{8}{20}

    Divide the new length by the old length — new over old, in that order.

  2. Simplify the fraction

    k=25k = \frac{2}{5}

    Cancelling gives the exact scale factor 25\frac{2}{5}. Leave it as a fraction rather than a rounded decimal.

  3. State the scale factor of the enlargement

    PQR→ABC:k=25PQR \rightarrow ABC: \quad k = \frac{2}{5}

    So the scale factor from triangle PQRPQR to triangle ABCABC is 25\frac{2}{5}.

Answer
k=25k = \frac{2}{5}
Question 5
2 markseasy
Triangle ABCABC is similar to triangle PQRPQR, with AA corresponding to PP, BB to QQ and CC to RR. AB=6AB = 6 cm, PQ=9PQ = 9 cm and BC=8BC = 8 cm. Work out the length of QRQR.

Worked solution

  1. Work out the scale factor from triangle ABC to triangle PQR

    k=PQAB=96=32k = \frac{PQ}{AB} = \frac{9}{6} = \frac{3}{2}

    Dividing the new length by the old one gives k=32k = \frac{3}{2}.

  2. Multiply the corresponding side by the scale factor

    QR=BC×k=8×32QR = BC \times k = 8 \times \frac{3}{2}

    QRQR corresponds to BCBC, so it is 32\frac{3}{2} times as long.

  3. Work out the missing length

    QR=12 cmQR = 12 \text{ cm}

    So QR=12QR = 12 cm.

Answer
QR=12 cmQR = 12 \text{ cm}

Unlock 65 more Similarity questions

Create a free account to work through every GCSE Similarity question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Similarity practice

Hard QuestionsPractice Questions

Related Geometry & Measures topics

3D solids, plans and elevations3D Pythagoras and trigonometryAngle factsArcs and sectorsBearings and scale drawingsCircle theorems