Match corresponding lengths in similar figures
Identify corresponding vertices and lengths · Use one common positive length scale factor · Reject similarity based only on appearance or one side
Section Paper 1 · Historical Core · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Ratio and proportion
- Triangle and quadrilateral definitions
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Identify corresponding vertices and lengths
- Use one common positive length scale factor
- Reject similarity based only on appearance or one side
Same shape, one scale
Similar figures have equal corresponding angles and lengths in one common positive ratio. Match corresponding vertices before comparing sides. A larger-to-smaller length factor k multiplies every corresponding length, including heights, but does not change angles.
For non-degenerate triangles, two equal corresponding angles establish the shape; alternatively, proportional corresponding side triples establish similarity. For general quadrilaterals, equal angles alone are insufficient: rectangles of different length-to-width ratios have different shapes.
Test all supplied information
Compare ratios in the same direction, not larger/smaller for one side and smaller/larger for another. If required corresponding side ratios differ, the figures are not similar. One matching side ratio by itself does not establish the whole shape.
Sources for this lesson
Pause and explain
Are rectangles 6 by 4 and 9 by 8 similar?
Worked example
A 3 by 4 rectangle is enlarged by length factor 1.5. Find its new side lengths.
Show the worked solution
- 3×1.5 = 4.5 and 4×1.5 = 6.
- Both lengths share the same factor; the right angles remain unchanged.
Answer 4.5 by 6, with unchanged corresponding angles.
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Test this topic →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 triangles have corresponding sides 4 and 10. What is the larger-to-smaller factor, and what corresponds to a smaller side 6?
Give me a hint
Use one ratio direction.
Compare my reasoning
- k = 10/4 = 2.5.
- 6×2.5 = 15.
Factor 2.5; corresponding larger side 15.
Look for these in your work
- Match corresponding sides
- Use the same direction throughout
Triangles with sides 3,4,5 and 6,8,10 are claimed similar. Check the corresponding side ratios.
Give me a hint
Compare the three ratios in order.
Compare my reasoning
- 6/3 = 8/4 = 10/5 = 2.
- All corresponding sides have the same factor, establishing triangle similarity.
Similar with length factor 2.
Look for these in your work
- Check all three sides
- Keep correspondence consistent
A poster 6 by 4 is resized to 9 by 8. Explain whether it keeps its shape and what a height preserving shape should be for width 9.
Give me a hint
Use the width factor for the height too.
Compare my reasoning
- Width factor 9/6 = 1.5 but height factor 8/4 = 2, so it is distorted.
- A shape-preserving height is 4×1.5 = 6.
Not shape-preserving; use height 6 for width 9.
Look for these in your work
- Test both dimensions
- Explain the corrected common scale
Common mistakes
- Match sides and ratio direction.
- Equal angles do not establish arbitrary-quadrilateral similarity on their own.
What can you explain now?
Are rectangles 6 by 4 and 9 by 8 similar?
Compare with the explanation
No; corresponding ratios 9/6 and 8/4 differ.
Their right angles match, but they lack one common length factor.
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Make it stick
Are rectangles 6 by 4 and 9 by 8 similar?
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.