Sheet № 59 · Foundation + Higher · AQA · Edexcel · OCR
Congruence and Similarity –
Congruence and similarity are fundamental geometry concepts tested at both Foundation and Higher tiers across AQA, Edexcel, and OCR GCSE Maths papers. Two shapes are congruent if they are exactly the same shape and size. Two shapes are similar if they are the same shape but different sizes — one is an enlargement of the other. You need to
§Key definitions
Question:
Two similar rectangles have widths of 4 cm and 10 cm. The smaller rectangle has a length of 6 cm. Find the length of the larger rectangle.
Answer:
The length of the larger rectangle is 15 cm.
Question 1:
Two similar triangles have corresponding sides of 6 cm and 9 cm. What is the linear scale factor?
Question 2:
Two similar shapes have a linear scale factor of 4. The area of the smaller shape is 20 cm². Find the area of the larger shape.
Question 3:
Two similar solids have volumes of 27 cm³ and 216 cm³. Find the linear scale factor.
§Formulas to memorise
Linear scale factor = new length / original length
Area scale factor = (linear scale factor)²
Volume scale factor = (linear scale factor)³
SSS — three pairs of equal sides.
SAS — two pairs of equal sides and the included angle between them is equal.
ASA — (or AAS) — two pairs of equal angles and a corresponding side is equal.
RHS — both triangles have a right angle, equal hypotenuses, and one other pair of equal sides.
List the equal pairs with reasons (e.g. "given", "common side", "vertically opposite angles").
Worked example
Two similar rectangles have widths of 4 cm and 10 cm. The smaller rectangle has a length of 6 cm. Find the length of the larger rectangle.
Working:
⚠ Common mistakes
- ✗Using the wrong congruence condition. SSA (two sides and a non-included angle) is not a valid condition for congruence. Only SSS, SAS, ASA/AAS, and RHS work.
- ✗Confusing congruence with similarity. Congruent shapes are the same size; similar shapes are the same shape but can be different sizes. All congruent shapes are similar, but not all similar shapes are congruent.
- ✗Using the linear scale factor for area. If the linear scale factor is 3, the area factor is 9, not 3. Always square for area and cube for volume.
- ✗Matching the wrong corresponding sides. When finding the scale factor, make sure you are comparing sides that are in the same position on each shape. The longest side on one shape corresponds to the longest side on the other.
- ✗Not simplifying the scale factor. A scale factor of 6/4 should be simplified to 3/2 or written as 1.5 for clarity.
✦ Exam tips
- →For congruence proofs, list your pairs of equal elements clearly and number them. Then state the condition: "By SAS, triangle ABC is congruent to triangle DEF."
- →Look for hidden similar triangles. When two triangles share an angle and have parallel sides, they are often similar. This is common in Higher tier questions involving parallel lines cutting a triangle.
- →Write the scale factor as a fraction if dividing gives a recurring decimal — this avoids rounding errors.
- →Always identify which shape is the "original" and which is the "new". This determines whether you multiply or divide by the scale factor.
- →Volume and area questions often appear in context — water tanks, paint coverage, model buildings. Read carefully to identify whether the question asks for length, area, or volume.