Class 11 Notes - Triangles

Download PDF Worksheets
Start a Practice Test

Triangles

In Class 11, the study of triangles deepens your understanding of geometry and trigonometry. You will explore the properties, types, and important theorems related to triangles, as well as their applications in solving real-world and mathematical problems.

1. Definition and Classification

  • Triangle: A polygon with three sides and three angles.
  • Types by Sides:
    • Equilateral Triangle (all sides equal)
    • Isosceles Triangle (two sides equal)
    • Scalene Triangle (all sides unequal)
  • Types by Angles:
    • Acute-angled Triangle (all angles < 90°)
    • Right-angled Triangle (one angle = 90°)
    • Obtuse-angled Triangle (one angle > 90°)

2. Properties of Triangles

  • The sum of the interior angles of a triangle is always 180°.
  • The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
  • In any triangle, the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality).

3. Congruence and Similarity

  • Congruent Triangles: Triangles that are identical in shape and size. Criteria: SSS, SAS, ASA, AAS, RHS.
  • Similar Triangles: Triangles with the same shape but not necessarily the same size. Criteria: AA, SSS, SAS.

4. Important Theorems

  • Pythagoras Theorem: In a right-angled triangle, \( a^2 + b^2 = c^2 \), where c is the hypotenuse.
  • Basic Proportionality Theorem (Thales’ Theorem): If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
  • Angle Bisector Theorem: The angle bisector divides the opposite side in the ratio of the adjacent sides.
  • Area Theorems: Area = \( \frac{1}{2} \times \text{base} \times \text{height} \); Heron's formula for area using side lengths.

5. Medians, Altitudes, and Centers

  • Medians: Line segments joining a vertex to the midpoint of the opposite side. All medians meet at the centroid.
  • Altitudes: Perpendiculars from a vertex to the opposite side. All altitudes meet at the orthocenter.
  • Angle Bisectors: Divide the angle into two equal parts. All meet at the incenter.
  • Perpendicular Bisectors: Bisect a side at 90°. All meet at the circumcenter.

6. Trigonometry in Triangles

  • Introduction to sine, cosine, and tangent ratios in right-angled triangles.
  • Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
  • Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos C \)

7. Example Problems

  1. Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
    Solution: Use Heron's formula.
    \( s = \frac{7+8+9}{2} = 12 \)
    \( \text{Area} = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.83 \) cm²
  2. In a right triangle, if one side is 6 cm and the hypotenuse is 10 cm, find the other side.
    Solution: \( a^2 + 6^2 = 10^2 \Rightarrow a^2 = 100 - 36 = 64 \Rightarrow a = 8 \) cm

8. Practice Exercises

  • Classify the following triangles by sides and angles: (a) 5 cm, 5 cm, 8 cm; (b) 7 cm, 8 cm, 9 cm.
  • Prove that the sum of the angles of a triangle is 180°.
  • Find the centroid of a triangle with vertices (0,0), (6,0), (0,6).
  • Solve for x: In triangle ABC, AB = AC, ∠B = 50°, find ∠C.
  • Find the area of an equilateral triangle with side 10 cm.

9. Summary

  • Triangles are fundamental shapes in geometry with many properties and theorems.
  • Understanding triangles is essential for higher mathematics, trigonometry, and real-world applications.