Triangles Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Introduction to Triangles

• A triangle is a polygon with three sides and three angles. It is one of the basic shapes in geometry and is classified based on side lengths and angle measures.

2. Classification of Triangles

• Based on Sides:

  • Scalene Triangle: All sides and angles are unequal.
  • Isosceles Triangle: Two sides are equal, and the angles opposite those sides are equal.
  • Equilateral Triangle: All three sides are equal, and each angle measures 60°.

• Based on Angles:

  • Acute Triangle: All angles are less than 90°.
  • Right Triangle: One angle is exactly 90°.
  • Obtuse Triangle: One angle is greater than 90°.

3. Similarity of Triangles

• Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in proportion.

4. Criteria for Similarity of Triangles

• AA (Angle-Angle) Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
• SSS (Side-Side-Side) Criterion: If the corresponding sides of two triangles are in proportion, then the triangles are similar.
• SAS (Side-Angle-Side) Criterion: If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are in proportion, the triangles are similar.

5. Basic Proportionality Theorem (Thales' Theorem)

• If a line is drawn parallel to one side of a triangle to intersect the other two sides, the other two sides are divided in the same ratio. $$\text{If } DE \parallel BC, \text{ then } \frac{AD}{DB} = \frac{AE}{EC}$$

6. Pythagoras Theorem

• In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. $$\text{In } \triangle ABC, \text{ if } \angle B = 90^\circ, \text{ then } AC^2 = AB^2 + BC^2$$ This theorem is a special case of the similarity criterion.

7. Converse of the Pythagoras Theorem

• If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, the angle opposite to the first side is a right angle.

8. Area of Similar Triangles

• The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. $$\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2 = \left(\frac{CA}{FD}\right)^2$$

9. Important Properties and Theorems

• Altitude of an Equilateral Triangle: The altitude bisects the base and the angle from which it is drawn.
• Median of a Triangle: The median bisects the opposite side.
• Angle Bisector Theorem: The angle bisector of a triangle divides the opposite side into segments proportional to the other two sides.

Formula Sheet

1. Pythagoras Theorem:

   $$AC^2 = AB^2 + BC^2 \quad (\text{In a right-angled triangle})$$

2. Basic Proportionality Theorem (Thales' Theorem):

   $$\frac{AD}{DB} = \frac{AE}{EC} \quad (\text{If } DE \parallel BC)$$

3. Criteria for Similarity:

   • AA Criterion: $\triangle ABC \sim \triangle DEF$ if $\angle A = \angle D$ and $\angle B = \angle E$.
   • SSS Criterion: $\triangle ABC \sim \triangle DEF$ if $\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}$.
   • SAS Criterion: $\triangle ABC \sim \triangle DEF$ if $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D$.

4. Area of Similar Triangles:

   $$\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2$$

5. Converse of Pythagoras Theorem:

   $$\text{If } AC^2 = AB^2 + BC^2, \text{ then } \angle B = 90^\circ$$

6. Angle Bisector Theorem:

   $$\frac{AB}{AC} = \frac{BD}{DC} \quad (\text{If } AD \text{ is the angle bisector of } \angle BAC)$$

7. Ratio of Areas of Similar Triangles:

   $$\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{\text{Corresponding Sides of } \triangle ABC}{\text{Corresponding Sides of } \triangle DEF}\right)^2$$

Examples for Practice

1. In a right-angled triangle, the lengths of the two legs are 6 cm and 8 cm. Find the length of the hypotenuse.
2. Prove that in an equilateral triangle, the altitudes are equal.
3. If a triangle has sides of lengths 5 cm, 12 cm, and 13 cm, prove that it is a right-angled triangle.
4. In a triangle, a line parallel to one side divides the other two sides into segments of length 4 cm and 6 cm. Find the ratio of the corresponding sides of the triangle.