Triangles

Class 10 Maths - Chapter 6 | NCERT Formula Sheet

1. Similar Figures

Two figures are similar if:
• They have same shape
• Not necessarily same size
• Corresponding angles are equal
• Corresponding sides are proportional

2. Similarity of Triangles

Two triangles are similar if:
1. Corresponding angles are equal (AAA)
2. Corresponding sides are proportional (SSS)
3. One angle equal & including sides proportional (SAS)
If ?ABC ~ ?PQR, then:
• ?A = ?P, ?B = ?Q, ?C = ?R
• AB/PQ = BC/QR = AC/PR = k (scale factor)

3. Basic Proportionality Theorem (Thales Theorem)

Theorem: If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio.
In ?ABC, if DE || BC, then:
AD/DB = AE/EC
Thales Theorem Diagram

4. Criteria for Similarity of Triangles

AAA Criterion

If ?A = ?P, ?B = ?Q, ?C = ?R
Then ?ABC ~ ?PQR

SSS Criterion

If AB/PQ = BC/QR = AC/PR
Then ?ABC ~ ?PQR

SAS Criterion

If ?A = ?P and
AB/PQ = AC/PR
Then ?ABC ~ ?PQR

5. Area of Similar Triangles

Theorem: The ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
If ?ABC ~ ?PQR, then:
Area(?ABC)/Area(?PQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)²

6. Pythagoras Theorem

Theorem: In a right triangle, square of hypotenuse equals sum of squares of other two sides.
In right ?ABC, right-angled at B:
AC² = AB² + BC²
Converse: If AC² = AB² + BC², then ?B = 90°

7. Important Results

  1. If two triangles are similar, then ratio of their:
    • Altitudes = Ratio of corresponding sides
    • Medians = Ratio of corresponding sides
    • Angle bisectors = Ratio of corresponding sides
    • Perimeters = Ratio of corresponding sides
  2. Mid-point Theorem: Line joining mid-points of two sides is parallel to third side and half of it
  3. In right triangle, midpoint of hypotenuse is equidistant from all three vertices

8. Similarity Theorems

Triangle Proportionality Theorem Converse

If AD/DB = AE/EC, then DE || BC

Right Triangle Similarity

In right ?ABC (?B=90°), BD?AC, then:
• ?ADB ~ ?ABC
• ?BDC ~ ?ABC
• ?ADB ~ ?BDC

9. Quick Reference Table

Similarity Ratio

k = AB/PQ = BC/QR = AC/PR

Area Ratio

Area ratio = k²

Thales Theorem

If DE||BC, AD/DB=AE/EC

10. Example (NCERT Style)

Problem: In ?ABC, DE||BC. If AD=4cm, DB=6cm, AE=5cm, find EC.

Solution:
By Thales Theorem: AD/DB = AE/EC
4/6 = 5/EC
EC = (5×6)/4 = 30/4 = 7.5cm
Problem: Two triangles have sides 6,8,10 and 9,12,15. Are they similar?

Solution:
Check ratio: 6/9=2/3, 8/12=2/3, 10/15=2/3
All ratios equal ? Triangles are similar (SSS criterion)