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Triangles - Revision Notes — Class 10 Mathematics

Quick revision notes for Triangles chapter including key points and important formulas.

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📌 Key Points

  • Similar triangles have proportional corresponding sides and equal corresponding angles
  • AA Similarity: If two angles of one triangle equal two angles of another, triangles are similar
  • SSS Similarity: If sides of one triangle are proportional to another, triangles are similar
  • SAS Similarity: If one angle is equal and including sides are proportional, triangles are similar
  • Basic Proportionality Theorem: If DE || BC in △ABC, then AD/DB = AE/EC
  • Converse of BPT: If AD/DB = AE/EC, then DE || BC
  • Pythagoras Theorem: In right triangle, c² = a² + b² (hypotenuse² = sum of squares of other sides)
  • Converse of Pythagoras Theorem: If a² + b² = c², the triangle is right-angled
  • Ratio of areas of similar triangles = (Ratio of corresponding sides)²
  • Ratio of perimeters of similar triangles = Ratio of corresponding sides
  • When altitude is drawn to hypotenuse in right triangle: CD² = AD × DB (geometric mean)
  • Each leg is geometric mean of hypotenuse and adjacent segment
  • Common Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25)
  • In similar triangles, corresponding altitudes are in same ratio as corresponding sides
  • Midpoint theorem: Line joining midpoints of two sides is parallel to third side and half its length
  • If triangles are similar with similarity ratio k, area ratio = k²
  • The angle bisector theorem relates segments created by angle bisector
  • Triangles with equal altitudes have areas proportional to their bases

📘 Important Definitions

Similar Triangles
Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
Congruent Triangles
Two triangles are congruent if they have the same size and shape (all sides and angles equal).
Basic Proportionality Theorem (BPT)
If a line drawn parallel to one side of a triangle intersects the other two sides, it divides them proportionally.
Pythagoras Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
Hypotenuse
The longest side of a right-angled triangle, opposite to the right angle.
Altitude
A perpendicular line segment from a vertex of a triangle to the opposite side (or its extension).
Median
A line segment joining a vertex of a triangle to the midpoint of the opposite side.
Pythagorean Triplet
A set of three positive integers a, b, c such that a² + b² = c², satisfying Pythagoras theorem.
Similarity Ratio
The constant ratio between corresponding sides of two similar triangles.
Geometric Mean
For numbers a and b, geometric mean = √(a×b). In right triangle with altitude to hypotenuse: altitude² = segment1 × segment2.

⚠️ Common Mistakes

✗ Wrong: Confusing similarity with congruence - similar triangles have proportional sides, congruent have equal sides

✓ Correct: Similar triangles: angles equal, sides proportional. Congruent triangles: angles and sides both equal.

✗ Wrong: Using wrong similarity criterion - mixing AA with AAA or applying SSS without checking all three sides

✓ Correct: Learn three criteria: AA, SSS, SAS. AAA is redundant (if 2 angles equal, 3rd is automatic).

✗ Wrong: Setting up BPT proportion incorrectly - mixing ratios from different sides

✓ Correct: BPT: AD/DB = AE/EC (from same side). Don't write AD/AE = DB/EC.

✗ Wrong: Forgetting to square the similarity ratio when comparing areas

✓ Correct: Area ratio = (Side ratio)². If sides 1:2, areas 1:4.

✗ Wrong: Not identifying the hypotenuse correctly in Pythagoras theorem

✓ Correct: Hypotenuse is always opposite the right angle (largest side). Formula: c² = a² + b².

✗ Wrong: Applying Pythagoras theorem to non-right triangles

✓ Correct: Pythagoras theorem only applies to right-angled triangles. Check for 90° angle.

✗ Wrong: Incorrectly calculating similarity ratio from given information

✓ Correct: Ratio = corresponding side of first triangle / corresponding side of second triangle. Be consistent.

✗ Wrong: Confusing altitude segments - AD × DB should equal CD² not AD × DC

✓ Correct: When altitude CD drawn to hypotenuse AB: CD² = AD × DB (altitude is geometric mean of segments).

📝 Exam Focus

These questions are frequently asked in CBSE exams:

Prove two triangles are similar using different criteria
5m
Apply BPT to find unknown lengths with parallel lines and proportions
3m
Use Pythagoras theorem to find sides, check if triangle is right-angled, find altitude to hypotenuse
3m
Calculate area using similarity (given area and similarity ratio, find new area)
3m
Combination problems using multiple theorems (BPT, similarity, Pythagoras)
5m
Prove properties like geometric mean altitude, midpoint theorem
5m
Identify and use Pythagorean triplets in problems
1m
Find altitudes and other lengths in similar triangles
3m
Solve real-world problems involving similar triangles
3m
State and apply all three similarity criteria correctly
1m

🎯 Last-Minute Recall

Close your eyes and try to recall: Key definitions, formulas, and 3 common mistakes. If you can recall 80% without looking, you're exam-ready!