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If ΔABC ∼ ΔPQR with area ratio Area(ΔABC)/Area(ΔPQR) = 9/16 and BC = 4.5 cm, what is the length of QR?

6.0 cm. Area ratio = (BC / QR)² ⇒ 9/16 = (4.5 / QR)² ⇒ 3/4 = 4.5 / QR ⇒ QR = (4 × 4.5)/3 = 18/3 = 6.0 cm.

Mathematics · Class 9-10 · CBSE

Similar Triangles: Thales Theorem, Criteria and Pythagoras Theorem

Master geometric similarity of triangles: Congruent vs Similar figures, Basic Proportionality Theorem (BPT / Thales Theorem), similarity criteria (AAA, SSS, SAS), ratio of areas theorem, and Pythagoras theorem.

1Congruence vs Similarity of Geometric Figures

Two geometric figures areCongruent(≅) if they have the same shape AND the same size. Two figures areSimilar(∼) if they have the same shape, but not necessarily the same size. All congruent figures are similar, but similar figures need not be congruent.

Congruent Triangles vs Similar Triangles

Property	Congruent Triangles (ΔABC ≅ ΔDEF)	Similar Triangles (ΔABC ∼ ΔDEF)
Corresponding Angles	Identical: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F	Identical: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Corresponding Sides	$\text{Equal lengths}\ : AB = DE, BC = EF, AC = DF$	$\text{Proportional}\ : AB/DE = BC/EF = AC/DF$
Areas Ratio	Exact equal area: Area(ΔABC) = Area(ΔDEF)	Ratio of areas = Square of ratio of sides = (AB/DE)²
Scale Factor (k)	$k = 1$	k can be any positive real number

Corresponding Angles
Congruent Triangles (ΔABC ≅ ΔDEF)
Identical: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Similar Triangles (ΔABC ∼ ΔDEF)
Identical: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Corresponding Sides
Congruent Triangles (ΔABC ≅ ΔDEF)
$\text{Equal lengths}\ : AB = DE, BC = EF, AC = DF$
Similar Triangles (ΔABC ∼ ΔDEF)
$\text{Proportional}\ : AB/DE = BC/EF = AC/DF$
Areas Ratio
Congruent Triangles (ΔABC ≅ ΔDEF)
Exact equal area: Area(ΔABC) = Area(ΔDEF)
Similar Triangles (ΔABC ∼ ΔDEF)
Ratio of areas = Square of ratio of sides = (AB/DE)²
Scale Factor (k)
Congruent Triangles (ΔABC ≅ ΔDEF)
$k = 1$
Similar Triangles (ΔABC ∼ ΔDEF)
k can be any positive real number

2Basic Proportionality Theorem (Thales Theorem)

Thales Theorem (BPT): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Basic Proportionality Theorem Ratio

\text{In } \Delta ABC, \text{ if } DE \parallel BC \implies \frac{AD}{DB} = \frac{AE}{EC}

Logical Proof Outline of Thales Theorem

1
1. Given: In ΔABC, line DE is parallel to side BC, intersecting AB at D and AC at E.
2
2. Construction: Join BE and CD. Draw perpendiculars EN ⊥ AB and DM ⊥ AC.
3
3. Area ratio on side AB: Area(ΔADE)/Area(ΔBDE) = [(1/2)·AD·EN] / [(1/2)·DB·EN] = AD / DB.
4
4. Area ratio on side AC: Area(ΔADE)/Area(ΔCDE) = [(1/2)·AE·DM] / [(1/2)·EC·DM] = AE / EC.
5
5. Equating: ΔBDE and ΔCDE share same base DE between same parallels DE and BC, so Area(ΔBDE) = Area(ΔCDE).
6
$6. \text{Conclusion}\ : AD / DB = AE / EC.$

3Criteria for Similarity of Triangles

The Three Formal Similarity Criteria

Criterion	Condition Required	Mathematical Expression
AAA (or AA) Similarity	If two angles of one triangle are equal to two angles of another triangle	If ∠A = ∠D and ∠B = ∠E ⇒ ΔABC ∼ ΔDEF
SSS Similarity	If all three corresponding sides of two triangles are proportional	If AB/DE = BC/EF = AC/DF ⇒ ΔABC ∼ ΔDEF
SAS Similarity	If one angle of a triangle is equal to one angle of another and the including sides are proportional	If ∠A = ∠D and AB/DE = AC/DF ⇒ ΔABC ∼ ΔDEF

AAA (or AA) Similarity
Condition Required
If two angles of one triangle are equal to two angles of another triangle
Mathematical Expression
If ∠A = ∠D and ∠B = ∠E ⇒ ΔABC ∼ ΔDEF
SSS Similarity
Condition Required
If all three corresponding sides of two triangles are proportional
Mathematical Expression
If AB/DE = BC/EF = AC/DF ⇒ ΔABC ∼ ΔDEF
SAS Similarity
Condition Required
If one angle of a triangle is equal to one angle of another and the including sides are proportional
Mathematical Expression
If ∠A = ∠D and AB/DE = AC/DF ⇒ ΔABC ∼ ΔDEF

4Exam Corner: Tips, Mistakes & Board Questions

Critical Board Exam Theorems to Remember

Ratio of areas of two similar triangles equals the ratio of squares of their corresponding sides, altitudes, medians, or angle bisectors: Area(Δ1)/Area(Δ2) = (Side1/Side2)² = (Altitude1/Altitude2)².
Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides: AC² = AB² + BC².
Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Exam question · CBSE Class 10 Board

In ΔABC, DE || BC such that AD = 3 cm, DB = 5 cm, and AE = 1.8 cm. What is the length of AC?

A.3.0 cm
B.4.8 cm
C.3.6 cm
D.5.4 cm

Remember

Similar triangles haveequal corresponding anglesandproportional corresponding sides.
Thales Theorem (BPT): Line parallel to one side divides other sides proportionally.
Similarity criteria includeAA, SSS, and SAS.
Ratio of areas of similar triangles =(Ratio of corresponding sides)².

Check yourself

Question 1
If ΔABC ∼ ΔPQR with area ratio Area(ΔABC)/Area(ΔPQR) = 9/16 and BC = 4.5 cm, what is the length of QR?