Triangles — Class 10 Mathematics
"Two triangles are similar if their shape is identical, even if their sizes differ — the algebra of geometry."
1. About the Chapter
Class 10 Triangles chapter focuses on similarity — when two triangles have the same shape but different sizes. The chapter covers:
- Similarity criteria (AA, SAS, SSS)
- Basic Proportionality Theorem (BPT or Thales)
- Pythagoras theorem with proof
- Areas of similar triangles
Foundation for trigonometry, coordinate geometry, and higher math.
2. Similar Figures
Definition
Two figures are SIMILAR if they have:
- Same shape
- Different (or same) sizes
Examples
- Two squares (always similar)
- Two equilateral triangles (always similar)
- Photo enlargement of itself
Similar vs Congruent
- Congruent: same shape AND same size (≅)
- Similar: same shape only (~)
3. Similarity of Triangles
Definition
Two triangles ABC and PQR are similar if:
- Corresponding angles are EQUAL
- Corresponding sides are PROPORTIONAL
Written: △ABC ~ △PQR
Means:
- ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R
- AB/PQ = BC/QR = CA/RP
Similarity Ratio (Scale Factor)
The constant ratio AB/PQ = BC/QR = CA/RP is called the scale factor.
4. Similarity Criteria
1. AA (Angle-Angle) Criterion
If TWO angles of one triangle equal corresponding two angles of another, triangles are similar.
(Third angle is automatically equal because sum is 180°.)
2. SAS (Side-Angle-Side) Similarity
If two pairs of corresponding sides are proportional AND included angle is equal, triangles are similar.
3. SSS (Side-Side-Side) Similarity
If three pairs of corresponding sides are proportional, triangles are similar.
5. Basic Proportionality Theorem (BPT) / Thales Theorem
Statement
If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, then the other two sides are divided in the same ratio.
Proof (Outline)
In △ABC, DE is parallel to BC, meeting AB at D and AC at E. Then AD/DB = AE/EC.
(Proof uses similar triangles: △ABC ~ △ADE by AA criterion, since DE || BC.)
Converse
If a line divides two sides of a triangle in the same ratio, it is PARALLEL to the third side.
6. Pythagoras Theorem (Detailed)
Statement
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
If △ABC has right angle at B: AC² = AB² + BC²
Indian Origin
Discovered in Sulba Sutras (~800 BCE) by Baudhayana — 300 years before Pythagoras. Indian textbooks now call it the Baudhayana-Pythagoras Theorem.
Proof (Using Similarity)
In right-angled △ABC (right angle at B), drop perpendicular BD from B to AC.
Then:
- △ABD ~ △ABC (AA similarity)
- △BDC ~ △ABC (AA similarity)
From similarity:
- AB²/AC = AD/AB → AB² = AD × AC
- BC²/AC = DC/BC → BC² = DC × AC
Adding:
- AB² + BC² = AC × (AD + DC) = AC × AC = AC²
Therefore AC² = AB² + BC².
Converse
If in a triangle, the square on one side equals the sum of squares on the other two sides, then the triangle has a right angle opposite the first side.
Pythagorean Triples
Integer sides satisfying a² + b² = c²:
- (3, 4, 5)
- (5, 12, 13)
- (8, 15, 17)
- (7, 24, 25)
- (9, 40, 41)
- (20, 21, 29)
7. Areas of Similar Triangles
Theorem
Ratio of areas of two similar triangles = ratio of squares of corresponding sides.
If △ABC ~ △PQR with sides in ratio k:1, then: Area(△ABC) / Area(△PQR) = k²
Proof Sketch
- Base AB to PQ ratio is k
- Heights are also in ratio k (since triangles similar)
- Area = ½ × base × height
- So Area ratio = k × k = k²
Examples
If two similar triangles have side ratio 2:3, area ratio is 4:9. If side ratio is 1:2, area ratio is 1:4.
8. Worked Examples
Example 1: AA similarity
In △ABC, ∠B = 90°, ∠A = 30°. In △PQR, ∠Q = 90°, ∠P = 30°. Are they similar?
- ∠B = ∠Q = 90°
- ∠A = ∠P = 30°
- (Third angle: ∠C = ∠R = 60°)
- By AA: △ABC ~ △PQR ✓
Example 2: BPT
In △ABC, DE || BC, where D on AB and E on AC. If AD = 3 cm, DB = 5 cm, AE = 4 cm, find AC.
- AD/DB = AE/EC
- 3/5 = 4/EC → EC = 20/3
- AC = AE + EC = 4 + 20/3 = 32/3 cm
Example 3: Pythagoras
A 15 m ladder leans against wall. Foot 9 m from wall. How high up?
- Hypotenuse² = 15² = 225
- Base² = 9² = 81
- Height² = 225 − 81 = 144
- Height = 12 m
Example 4: Areas of similar
Triangles ABC and PQR are similar. Side AB = 4 cm, PQ = 6 cm. Area of △ABC = 24 cm². Find area of △PQR.
- Side ratio = 4:6 = 2:3
- Area ratio = (2/3)² = 4/9
- Area of △PQR = 24 × (9/4) = 54 cm²
Example 5: Pythagoras Triple Detection
Is the triangle with sides 9, 12, 15 a right triangle?
- Largest = 15. Check 9² + 12² = 81 + 144 = 225 = 15² ✓
- YES, right angle opposite side 15.
9. Common Mistakes
-
Similar = Congruent
- SIMILAR: same shape, possibly different sizes. CONGRUENT: same shape AND size.
-
Wrong correspondence in similarity
- △ABC ~ △PQR means: A↔P, B↔Q, C↔R. Maintain correspondence.
-
Area ratio = side ratio (forgot to square)
- Area ratio = (side ratio)². If sides 2:3, areas are 4:9.
-
Identifying hypotenuse wrong
- Hypotenuse is opposite the RIGHT angle, longest side.
-
Pythagoras for non-right triangles
- Theorem applies ONLY to right-angled triangles.
10. Real-World Applications
Surveying and Architecture
- Calculating heights of mountains using similar triangles (shadow method)
- Indian temples used similar triangle ratios
- Modern surveying uses theodolites based on these principles
Engineering
- Bridge truss design
- Roofing slopes
- Cable lengths in suspension bridges
Photography and Maps
- Scale models (similar to real objects)
- Map-to-ground distances
- Lens optics
11. Indian Heritage
- Baudhayana (~800 BCE): Sulba Sutras — Pythagoras theorem
- Aryabhata (5th century): triangle calculations
- Brahmagupta (7th century): cyclic quadrilaterals (extension)
- Bhaskara II (12th century): triangle area formulas
12. Conclusion
Triangles — especially similar triangles and Pythagoras theorem — are foundational to:
- Trigonometry (Chapters 8-9)
- Coordinate geometry (Chapter 7)
- 3D geometry
- Engineering and physics
Master:
- 3 similarity criteria
- BPT (Thales)
- Pythagoras theorem with proof
- Areas of similar triangles
Practice 15+ problems. Triangles are the geometry of our world.
