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Similarity

Similar shapes: lengths

Similarity Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Similar shapes and the scale factor
  4. Nested triangles and the X-shape
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 4.4: similar shapes - lengths
  • 4.4Know that similar shapes have equal angles and all lengths multiplied by one scale factor.
  • 4.4Show that two triangles are similar (two pairs of equal angles).
  • 4.4Find missing lengths, including nested triangles and the 'X-shape'.

Before you start: Angles in parallel lines and ratio. Next: Areas and volumes.

Formula bank

Scale factorLearn it\(k=\dfrac{\text{length on the new shape}}{\text{matching length on the old shape}}\)every length × k; angles stay the same.Use it when two similar shapes.
Similar trianglesLearn ittwo pairs of equal anglesthe third pair then matches too.Use it when showing triangles are similar.

Similar shapes and the scale factor

Similar shapes are enlargements of each other: same angles, every length multiplied by the same scale factor \(k\). A 3 cm by 5 cm rectangle and a 4.5 cm by 7.5 cm rectangle are similar (\(k=1.5\)); a 3 by 5 and a 4 by 6 are not (\(\tfrac43\ne\tfrac65\)).

General methodHow to find a missing length in similar shapes

  1. Match up the corresponding sides (same angles opposite them). Redraw the two shapes the same way round if needed.
  2. k = known length on the shape you want ÷ matching length on the other shape.
  3. Multiply (or divide) the matching length by k.
  4. Check: bigger shape → bigger length.

Example Triangle PQR is similar to triangle XYZ. PQ = 6 cm, QR = 8 cm, XY = 9 cm. Find YZ.

PQ ↔ XY gives k.

  1. \(k=9\div6=1.5\)
  2. \(YZ=8\times1.5\)

Answer 12 cm

Try it yourself

1
Scale factor from a side of 4 cm to the matching side of 10 cm?
Show answer

Answer 2.5

2
With k = 2.5, a side of 7 cm becomes? (cm)
Show answer

Answer 17.5

Nested triangles and the X-shape

Nested similar trianglesABCDE466?
DE is parallel to BC.

Angle A is shared and the parallel lines give equal corresponding angles, so triangles ADE and ABC are similar. Use WHOLE sides: AB = 4 + 6 = 10, so \(k=\tfrac{10}4=2.5\) and BC = 2.5 × 6 = 15.

X-shape: AB parallel to CDABCDX
AB is parallel to CD; AD and BC cross at X.

Vertically opposite angles at X and alternate angles at A and D make triangles ABX and DCX similar - with A ↔ D and B ↔ C. If AB = 4 and CD = 6, then \(k=1.5\).

General methodHow to work with nested triangles or an X-shape

  1. Give the two angle reasons (shared/vertically opposite angle + corresponding/alternate angles).
  2. Nested: use the WHOLE side of the big triangle (small part + extra part).
  3. X-shape: match the vertices across X carefully - the small triangle is upside down.
  4. Then find k and the missing length as usual.

Try it yourself

1
DE ∥ BC, AD = 3 cm, DB = 5 cm, DE = 4.5 cm. Find BC (cm).
Hint
AB = 8, so k = 8/3.
Show answer

Answer 12

Ways they test it

Type 1 Matching sides

"Triangle PQR is similar to triangle XYZ ... Find YZ." (2 marks)

Spot it 'similar' and side lengths.

  1. Match sides in letter order.
  2. k, then multiply.

Type 2 Nested triangles

"DE is parallel to BC. AD = 4, DB = 6, DE = 6. Find BC." (2 marks)

Spot it a small triangle inside a big one.

  1. Whole side AB = 10.
  2. k = 2.5.

For example k = 10 ÷ 4

    Answer BC = 15

    Type 3 X-shape: show similar and find

    "AB ∥ CD; AD meets BC at X. Show ABX and DCX are similar and find AB." (3 marks)

    Spot it two triangles meeting at a point.

    1. Two angle reasons.
    2. Match A ↔ D, B ↔ C; find k.

    For example AX = 4, XD = 10, CD = 12

    1. k = 2.5

    Answer AB = 4.8

    Mistakes to avoid

    • ✗ Using the extra piece DB in a nested triangle✓ the whole side ABThe triangles share the vertex A.
    • ✗ Adding a constant✓ multiply by kAn enlargement scales lengths.
    • ✗ Matching by position on the page✓ match by equal anglesX-shapes are flipped.

    Summary

    • Similar: equal angles, every length × k.
    • k = new length ÷ matching old length.
    • Triangles: two pairs of equal angles is enough.
    • Nested: use whole sides. X-shape: match vertices carefully.

    If you see …

    • a line parallel to a side of a trianglenested similar triangles.
    • two lines crossing between parallel linesan X-shape: similar triangles.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Rectangles 3 by 5 and 4.5 by 7.5. Scale factor?
    Show answer

    Answer 1.5

    2
    PQR ~ XYZ, PQ = 6, XY = 9, PR = 7. Find XZ.
    Show answer

    Answer 10.5

    3
    Nested: AD = 2, AB = 6, DE = 3 (DE ∥ BC). Find BC.
    Show answer

    Answer 9

    Medium

    1
    DE ∥ BC, AD = 4 cm, DB = 6 cm, DE = 6 cm. Find BC (cm).
    Guide me

    What is it asking? a length

    What do we know? AB = 10

    Plan k = 2.5

    Show working
    1. 6 × 2.5 = 15

    Answer 15

    2
    AB ∥ CD; AD and BC cross at X. AB = 4, CD = 6, BX = 3.6. Find CX.
    Guide me

    What is it asking? a length

    What do we know? k = 6 ÷ 4 = 1.5

    Plan B ↔ C

    Show working
    1. 3.6 × 1.5 = 5.4

    Answer 5.4

    3
    In triangle ABC, P is on AB and Q on AC with PQ ∥ BC. AP = 5, PB = 3, BC = 12. Find PQ.
    Guide me

    What is it asking? a length

    What do we know? AB = 8

    Plan \(k=\tfrac58\)

    Show working
    1. 12 × 5/8 = 7.5

    Answer 7.5

    Hard

    1
    AB ∥ CD; AD meets BC at X. AX = 4 cm, XD = 10 cm, CD = 12 cm. Find AB (cm).
    Guide me

    What is it asking? a length

    What do we know? A ↔ D: k = 10 ÷ 4 = 2.5

    Plan AB = CD ÷ 2.5

    Show working
    1. 12 ÷ 2.5 = 4.8

    Answer 4.8

    PartAnswer / workingMarksGuidance
    angle reasonsB1
    k = 2.5M1
    4.8A1
    Total: 3 marks
    2
    DE ∥ BC, AD = 3 cm, DB = 5 cm, DE = 4.5 cm. Find BC (cm).
    Guide me

    What is it asking? a length

    What do we know? AB = 8

    Plan k = 8/3

    Show working
    1. 4.5 × 8/3 = 12

    Answer 12

    PartAnswer / workingMarksGuidance
    k = 8/3M1
    12A1
    Total: 2 marks
    3
    A 6 cm by 4 cm rectangle is enlarged so its short side is 10 cm. Find the long side (cm).
    Guide me

    What is it asking? a length

    What do we know? k = 10 ÷ 4

    Plan 6 × 2.5

    Show working
    1. 15

    Answer 15

    PartAnswer / workingMarksGuidance
    k = 2.5M1
    15A1
    Total: 2 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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