Similar lengths and matching triangles
Match corresponding sides, use one scale factor and justify triangle similarity when required.
Core and Extended: corresponding lengths. Extended also requires showing triangles are similar with geometric reasons; the angle justification below is labelled accordingly.
Before you begin
- Simplify ratios.
- Use triangle angle sums.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What multiplier changes 4 into 6?
What you will learn
- Find corresponding lengths in similar shapes.
- Use a scale factor in the correct direction.
- For Extended, establish triangle similarity from matching angles.
Correspondence comes before the ratio
Similar shapes have equal corresponding angles and one common multiplier for all corresponding lengths. For triangle ABC similar to triangle DEF, the stated order matches A with D, B with E and C with F. Thus AB corresponds to DE and BC to EF. The scale factor from ABC to DEF is DE ÷ AB, using new length over original length.
If AB = 4 cm and DE = 6 cm, the enlargement factor is 1.5. A corresponding side BC = 8 cm becomes EF = 12 cm. Going back uses the reciprocal factor 2/3. Shape orientation does not change correspondence: triangles can be turned or reflected, so match angles or stated vertex order rather than choosing the visually nearest side.
Extended: explain why triangles are similar
Two pairs of equal corresponding angles establish similarity, often called AA. The third pair then matches because each triangle's angles total 180°. State the actual angle equalities and their reasons, such as alternate angles in parallel lines and a shared angle; writing only 'they look similar' is insufficient.
If D lies on AB, E on AC and DE is parallel to BC, triangle ADE shares angle A with ABC. Their angles at D and B are corresponding angles on parallel lines, so the triangles are similar. Corresponding sides AD/AB, AE/AC and DE/BC have the same ratio. These are whole-to-whole comparisons: do not replace AB with the remaining part DB.
Pause and explain
Similar triangles have AB ↔ DE and BC ↔ EF. Which ratios match?
Worked example
Triangles ABC and DEF are similar in that order. AB = 4 cm, DE = 6 cm and BC = 8 cm. Find EF.
Show the worked solution
- Match AB to DE and BC to EF from the vertex order.
- The factor from ABC to DEF is 6 ÷ 4 = 1.5.
- EF = 8 × 1.5 = 12 cm; both side pairs have the same ratio.
Answer EF = 12 cm.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Similar shapes have corresponding sides 5 cm and 15 cm. A second side of the smaller shape is 7 cm. Find the larger side.
Give me a hint
Use the larger-over-smaller multiplier.
Compare my reasoning
- The scale factor is 15 ÷ 5 = 3.
- Multiply the smaller second side: 7 × 3 = 21 cm.
- Check 21/7 = 15/5 = 3.
21 cm.
Look for these in your work
- Matched corresponding sides.
- Used one factor for every length.
A large similar triangle has sides corresponding to 18 cm and 12 cm in the smaller one. Another large side is 27 cm. Find its smaller partner.
Give me a hint
The requested direction is large to small.
Compare my reasoning
- The reduction factor is 12/18 = 2/3.
- Multiply 27 by 2/3 to obtain 18 cm.
- The ratios small/large are 12/18 and 18/27, both 2/3.
18 cm.
Look for these in your work
- Used the reduction factor.
- Checked the common ratio.
Extended: D lies on AB and E on AC, with DE parallel to BC. AD = 6 cm, AB = 10 cm and BC = 15 cm. Justify similarity and find DE.
Give me a hint
Use the shared angle and a pair of corresponding angles.
Compare my reasoning
- Angle DAE equals BAC, and ADE equals ABC by corresponding angles, so ADE and ABC are similar by AA.
- The small-to-large ratio is AD/AB = 6/10 = 3/5.
- DE = 15 × 3/5 = 9 cm; compare AD with the whole AB.
Triangles ADE and ABC are similar by AA; DE = 9 cm (Extended justification).
Look for these in your work
- Gave two angle equalities and reasons.
- Used whole corresponding sides rather than DB.
Common mistakes
- Adding a difference instead of multiplying by a scale factor.
- Mixing directions in a proportion.
- For Extended, claiming similarity solely from appearance.
What can you explain now?
An enlargement multiplies all lengths by 2.5. What factor returns to the original?
Compare with the explanation
0.4
The return factor is the reciprocal, 1/2.5, so the two multipliers have product one.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, turn one similar triangle upside down. Match vertices before finding a missing side and, for Extended, state two angle reasons.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources