Lecture 9: Geometry - Triangle Formation Conditions

Key Topics Covered

• Conditions for triangle formation
• Types of triangles
• Conditions for specific triangle types (right-angled, obtuse, acute)
• Examples and questions to understand the concepts better

Triangle Formation Condition

• When two sides are given, the third side should always be:
  • Less than the sum of the two sides
  • Greater than the difference of the two sides
• Simply stated: Sum of any two sides is greater than the third side.

Example

• Given sides: 7 and 12
• To find integral values of third side (n):
  • n must be less than 19 and greater than 5
  • n ranges from 6 to 18
  • Total values: 13 (both inclusive)

Obtuse, Acute, and Right-Angled Triangles

Conditions for Different Types of Triangles

• Obtuse Angle Triangle: One angle > 90°
  • Condition: Longest side squared > sum of squares of other two sides.
  • Example: 12, 33, 7.62
• Acute Angle Triangle: All angles < 90°
  • Condition: Longest side squared < sum of squares of other two sides
• Right-Angle Triangle: One angle = 90°
  • Condition: Longest side squared = sum of squares of other two sides (Pythagorean theorem)

Examples to Identify Triangle Types

• Triangles with sides 12, 18, 23, Identifying Type:
  • Using conditions for obtuse angle triangle: Correct, it forms an obtuse triangle
• Triangles with sides 5, 12, 14:
  • Condition check, conclusion: acute angle triangle

Additional Concepts

Perimeter and Triangle Formation

• Formation of triangles when the perimeter (sum of all three sides) is given will be discussed in next lecture.

Practice Problems

1. Problem 1: Given sides 8 and 15, number of obtuse angle triangles

2. Solution:

   • Total values of third side from 8 to 22 = 15
   • Determine obtuse angle triangles:
     • Case 1: X < 15
     • Case 2: X > 15
   • Conclusion: Total obtuse angle triangles = 10
   • Number of acute angle triangles + right-angle triangle also derived.

3. Problem 2: Given sides 10 and 24, find number of acute angle triangles for third side

   • Using conditions for acute triangles to solve the problem.
   • Conclusion: The third side can take 4 integral values.

Note: Practicing such questions helps in solidifying the understanding of these concepts.