**CHAPTER 6: TRIANGLES — SIMILARITY OF FIGURES**
**DEFINITION OF SIMILAR FIGURES**
• Two figures are **similar** if they have the **same shape but not necessarily the same size**
• Two figures are **congruent** if they have the **same shape and same size**
• Key relationship: **All congruent figures are similar, but all similar figures are NOT congruent**
• Examples of similar figures: All circles are similar to each other; All squares are similar to each other; All equilateral triangles are similar to each other
• Different shapes (circle and square) are NOT similar — similarity requires same shape
**DEFINITION OF SIMILAR POLYGONS**
• Two polygons with the **same number of sides** are similar if and only if:
• **Both conditions MUST be satisfied** — either condition alone is insufficient
• If only angles are equal but sides are not proportional → NOT similar (e.g., square and rectangle)
• If only sides are proportional but angles are not equal → NOT similar (e.g., square and rhombus)
• **Scale Factor (Representative Fraction)**: The common ratio of corresponding sides
**NOTATION AND CORRESPONDENCE**
• Symbol **~** means "is similar to"
• When writing ABCD ~ PQRS, the **order of vertices matters** — it indicates which vertices correspond
• Correspondence: A ↔ P, B ↔ Q, C ↔ R, D ↔ S
• Corresponding angles: ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R, ∠D = ∠S
• Corresponding sides: AB/PQ = BC/QR = CD/RS = DA/SP = k (scale factor)
**PRACTICAL APPLICATIONS OF SIMILARITY**
• **Photography and enlargement**: Photographs of different sizes from the same negative are similar — each linear dimension is enlarged/reduced by the same scale factor
• **Indirect measurement**: Used to find heights of mountains, distances to celestial objects (moon, sun) without direct measurement
• **Maps and blueprints**: Drawn using appropriate scale factors (e.g., 1 cm = 1 km on a map)
• All angles remain **unchanged** during enlargement/reduction — only linear dimensions change by scale factor
**KEY CONCEPT: ACTIVITY — LIGHT AND SHADOW**
• When a cardboard cutout of polygon ABCD is placed between a light source O and a table, its shadow ABCD on the table is similar to original ABCD
• This is because light travels in straight lines → A lies on ray OA, B lies on ray OB, etc.
• The shadow is an **enlargement** (magnification) of the original
• Both conditions satisfied: equal corresponding angles + sides in same ratio
**IMPORTANT PROPERTIES**
• **Transitivity of similarity**: If polygon A ~ polygon B AND polygon B ~ polygon C, THEN polygon A ~ polygon C
• **Reflexivity**: Every polygon is similar to itself (trivial case with scale factor k=1)
• **Symmetry**: If polygon A ~ polygon B, then polygon B ~ polygon A
• **Scale factor relationship**: If scale factor from figure 1 to figure 2 is k, then scale factor from figure 2 to figure 1 is 1/k
**COMMON MISTAKES TO AVOID**
• ❌ Assuming two figures are similar if they have equal angles — corresponding sides must ALSO be proportional
• ❌ Assuming two figures are similar if corresponding sides are proportional — corresponding angles must ALSO be equal
• ❌ Confusing similarity with congruence — congruent means same shape AND size; similar means same shape only
• ❌ Not maintaining correct order of vertices when writing similarity statement — order indicates which vertices correspond
• ❌ Forgetting that scale factor is the ratio of corresponding sides, not individual measurements
• ❌ Assuming all rectangles are similar — only rectangles with proportional dimensions are similar
• ❌ Assuming all quadrilaterals with one right angle are similar — shape must be completely identical
**PROBLEM-SOLVING APPROACH**
**To determine if two polygons are similar:**
• Step 1: Check if they have same number of sides
• Step 2: Verify all corresponding angles are equal (use angle measurement or geometric properties)
• Step 3: Calculate ratios of all corresponding sides
• Step 4: Check if all ratios are equal (the scale factor)
• Step 5: If BOTH conditions satisfied → Similar; if either fails → NOT similar
**To find unknown sides when similarity is known:**
• Establish correspondence between vertices
• Write proportion: Side₁/Corresponding Side₂ = Scale Factor
• Use cross-multiplication to solve for unknown
**EXAMINATION TIPS**
• Always write similarity statement with correct vertex order: △ABC ~ △PQR (not △ABC ~ △RQP)
• State both conditions explicitly when proving similarity
• For quadrilaterals/polygons, verify correspondence clearly before checking ratios
• Draw clear diagrams showing corresponding parts
• Use similarity for indirect measurements — state the proportion clearly
• Remember: Equal angles alone ≠ Similar; Proportional sides alone ≠ Similar (both needed)
Q1. A photographer enlarges a 35mm photograph to 70mm size. If a line segment in the original photograph measures 5cm, what can be said about the corresponding line segment in the enlarged photograph using the concept of similar figures?
Answer: A — Similar figures have corresponding sides in the same ratio; the scale factor 35:70 = 1:2 means all linear dimensions double, independent of angles being equal (which are automatically equal in photographs of the same object).
Q2. Two quadrilaterals have all four corresponding angles equal. A student claims they must be similar. Is this claim justified based on the definition of similarity of polygons given in the chapter?
Answer: B — The chapter explicitly states similarity requires two conditions: (i) equal corresponding angles AND (ii) corresponding sides in the same ratio; a square and rectangle have equal corresponding angles but unequal side ratios, so they are not similar.
Q3. Assertion (A): All circles are similar to each other. Reason (R): All circles have the same shape but not necessarily the same size. Choose the correct option:
Answer: A — Both statements are true; Reason R explains Assertion A because the definition of similar figures explicitly requires same shape (not necessarily same size), which is precisely why all circles of different radii are similar.
Q4. A student observes that all congruent figures are similar. Based on the chapter's definitions, can the converse also be true—that all similar figures are congruent?
Answer: B — The chapter explicitly states 'all congruent figures are similar but similar figures need not be congruent'; congruence requires identical size and shape, while similarity only requires the same shape.
Q5. Assertion (A): A rectangle and a square cannot be similar. Reason (R): They have different numbers of sides. Choose the correct option:
Answer: C — Assertion A is true (rectangles and squares are not similar because side ratios differ), but Reason R is false (both have 4 sides); the correct reason is that corresponding sides are not in the same ratio, as shown in Fig. 6.6.
Q6. In Activity 1 of the chapter, a cardboard quadrilateral ABCD is placed between a bulb and a table to cast a shadow ABCD on the table. What property of light and geometry ensures that the shadow is similar to the original quadrilateral?
Answer: A — Light's straight-line propagation causes the shadow to be a magnified (or scaled) version of the original with corresponding vertices collinear with the light source, preserving both angles and side ratios—the exact conditions for similarity.
Q7. Assertion (A): If the scale factor between two similar polygons is 2:1, then each side of the first polygon is twice the corresponding side of the second polygon. Reason (R): The scale factor represents the ratio of corresponding sides in similar figures. Choose the correct option:
Answer: A — Both A and R are true; the scale factor 2:1 directly means corresponding sides have ratio 2:1, making sides of the first polygon twice those of the second—R correctly explains why A is true.
Q8. Two figures have the same shape but one is a 3D solid and the other is a 2D drawing. Can these figures be similar according to the definition given in the chapter?
Answer: B — The chapter defines similarity for polygons (2D figures) based on corresponding angles and sides; comparing a 3D solid to a 2D figure violates this definition because they are not comparable geometric objects.
Q9. A student claims: 'If polygon P is similar to polygon Q with scale factor 3:1, and polygon Q is similar to polygon R with scale factor 2:1, then polygon P is similar to polygon R with scale factor 6:1.' Is this reasoning correct?
Answer: A — The chapter's remark explicitly states that if one polygon is similar to another and that polygon is similar to a third, then the first is similar to the third; composing scale factors 3:1 and 2:1 gives 6:1.
Q10. Assertion (A): Two quadrilaterals with all corresponding sides in the ratio 1:1 must be congruent. Reason (R): When corresponding sides are in the ratio 1:1, all sides of both quadrilaterals are equal in length. Choose the correct option:
Answer: B — Both A and R are true (equal ratios mean equal side lengths), but this alone does not guarantee congruence unless angles are also equal; R describes a necessary but not sufficient condition for the conclusion in A.
What is the definition of similar figures?
Two figures are similar if they have the same shape but not necessarily the same size.
State the two conditions for two polygons to be similar.
Their corresponding angles must be equal AND their corresponding sides must be in the same ratio (proportion).
Are all congruent figures similar? Explain.
Yes, all congruent figures are similar because they have both the same shape and same size, which satisfies the similarity conditions.
Are all similar figures congruent?
No, similar figures need not be congruent because they can have different sizes.
Define scale factor (Representative Fraction).
The scale factor is the constant ratio between corresponding sides of two similar polygons.
Can a square and a rectangle be similar? Why or why not?
No, because although their corresponding angles are equal (all 90°), their corresponding sides are not in the same ratio.
Can a square and a rhombus be similar? Why or why not?
No, because although their corresponding sides are in the same ratio, their corresponding angles are not equal.
Are all circles similar to each other?
Yes, all circles are similar because they have the same shape regardless of their radius.
Are all equilateral triangles similar to each other?
Yes, all equilateral triangles are similar because they have all angles equal (60° each) and sides in the same ratio.
What real-life application uses the concept of similar figures?
Photographers enlarge or reduce photographs using a fixed scale factor to maintain the same shape at different sizes.
Define similar figures. Give one example of a pair of similar figures and one example of a pair of non-similar figures. [2 marks]
State that similar figures have same shape but not necessarily same size. For examples: similar pair could be two circles of different radii OR two equilateral triangles of different side lengths; non-similar pair could be a square and rectangle OR a circle and triangle.
Two quadrilaterals ABCD and PQRS are similar with a scale factor of 1:3. If AB = 5 cm, BC = 7 cm, and CD = 6 cm, find the corresponding sides PQ, QR, and RS of quadrilateral PQRS. Justify your answer. [3 marks]
Use the definition that corresponding sides of similar polygons are in the same ratio (scale factor). If scale factor is 1:3, then each side of PQRS = 3 × corresponding side of ABCD. Apply this formula to each side: PQ = 3 × AB = 15 cm, QR = 3 × BC = 21 cm, RS = 3 × CD = 18 cm.
Explain with reasons why a square and a rectangle are NOT similar figures, even though they appear to have related properties. Use a specific example with measurements to support your explanation. [5 marks]
State that similarity requires BOTH conditions: (1) corresponding angles equal AND (2) corresponding sides in same ratio. Show that while a square and rectangle both have all 90° angles (condition 1 satisfied), their sides are NOT in the same ratio (condition 2 fails). Use example: square with all sides 4 cm vs rectangle with sides 4 cm and 8 cm—angles are equal but side ratios are 4:4 vs 4:8, which are different. Conclude that having only one condition satisfied is insufficient for similarity.
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