Lecture Notes on Geometry

Introduction

• Lecturer: Ravi Prakash
• Topic: Geometry, focusing on triangles and specific problems around them.

Key Concepts

Right Angle Triangle Problems

• Problem: Given sides of a triangle are 15, 20, and 25. Need to find the sum of in-radii of triangles ABD and ADB.
  • Sides: 15, 20, 25 (Pythagorean triplet)
  • Hypotenuse: 25
  • Use semi-perimeter minus hypotenuse to find in-radii.
  • Height calculated using area formula.
  • Area = 150, height = 12 (calculated using ½ * base * height).
  • Using Pythagorean theorem, BD = 9.
  • In-radii calculated: r1 = 3, r2 = 4.
  • Sum of in-radii: R1 + R2 = 7.

Second Problem

• Problem: Find the distance between in-centers of triangles formed in a rectangle (sides = 9, 12).
  • Key concept: Pythagorean Theorem for finding lengths of non-horizontal/non-vertical lines.
  • Concept of circles in a rectangle moving apart as the rectangle stretches.
  • Solved using summed distances and radii, applying the Pythagorean Theorem to find final lengths.
  • Distance = 3√5.

Third Problem

• Problem: In a right-angled triangle, area = 80 square units, and perimeter = 80 units. Find hypotenuse length.
  • Key formula: In-radius = semi-perimeter - hypotenuse.
  • Use area = R * S to find in-radius.
  • Hypotenuse calculated as 38 units.

Fourth Problem

• Problem: Find the area of a triangle with circumradius = 18, in-radius = 8.
  • Key formulas: Semi-perimeter minus hypotenuse, area = R * S.
  • Calculated area: 352.

Fifth Problem

• Problem: In a right-angled triangle, if 2A + 7C = 9B and A = 12, find C.
  • Utilizing Pythagorean triplets and relationships between sides.
  • Found A, B, C triplet fitting given conditions.
  • Correct value of C = 25.5.

Important Formulas and Concepts

• Semi-perimeter (s): (Sum of all sides) / 2.
• In-radius (r): (Area of the triangle) / (Semi-perimeter)
• Pythagorean Triplets: Sets of three integers that satisfy a² + b² = c², useful in solving triangle sides.
• Area of Triangle: 0.5 * base * height or (in-radius * semi-perimeter).
• Distance between points: Using Pythagorean theorem for non-horizontal/vertical lines: √(Δx² + Δy²).

Conclusion

• The problems discussed primarily deal with properties of right-angled triangles and leverage Pythagorean identities, area calculations, and geometric properties of circles.
• Remember fundamental formulas and properties like in-radius, semi-perimeter, area calculations, and triplets for solving related geometric problems.