Six Plus or Minus Geometry Lecture by Ravi Prakash

Jul 16, 2024

Six Plus or Minus Geometry Lecture by Ravi Prakash

1. Introduction

  • Focus: Geometry problems, specifically involving triangles
  • Recap: Previous lessons on writing triangle, Pythagorean triplets

2. Example Problems and Solutions

Problem 1: Right-Angle Triangle with Sides 15, 20, 25

  1. Triangle Sides: 15, 20, 25 (Reduces to a 3-4-5 triangle)
    • Determine it is a right-angle triangle
  2. Goal: Find the sum of inradii of triangles ABD and ACD

Steps:

  1. Identify hypotenuse (25), making it the right-angle triangle.
  2. Calculate area using the formula: Area = 1/2 × base × height = 1/2 × 15 × 20 = 150.
  3. Calculate height AD using Area = 1/2 × height × hypotenuse: Height = 12.
  4. Using Pythagoras theorem: BD = √(AB² - AD²) = √(15² - 12²) = 9
  5. For inradius (r1 and r2): r1 = (semi-perimeter - hypotenuse) = 3, r2 = (semi-perimeter - hypotenuse) = 4
  6. Answer: Sum of inradii = r1 + r2 = 7.

Problem 2: Distance Between Incentre of Triangles in Rectangle ABCD

Given: Rectangle ABCD with sides 9 and 12. Goal: Find the distance between the incentres of triangles ABC and BCD.

Steps:

  1. Apply Pythagorean theorem: Diagonal of rectangle = 15.
  2. Recognize that radii of incircles of both triangles will not coincide due to the rectangular shape.
  3. Use distance formula decomposed into horizontal and vertical distances.
  4. Perpendicular and horizontal calculations indicate radii will sum appropriately.
  5. Answer: Distance PQ = 3√5.

Problem 3: Right-Angled Triangle with Specific Area and Perimeter

Given: Area = 80 sq units, Perimeter = 80 units Goal: Find length of hypotenuse.

Steps:

  1. Use formulas Area = R × S and in-radius = semi-perimeter - hypotenuse
  2. Calculate semi-perimeter: 40.
  3. Solve: Hypotenuse = 38.

Problem 4: Triangle with Given Circumradius and Inradius

Given: Circumradius = 18, Inradius = 8 Goal: Find the area of the triangle.

Steps:

  1. Formula: Hypotenuse = 36
  2. Calculate semi-perimeter: 44; perimeter: 88.
  3. Area calculation: Area = R × S = 352.
  4. Answer: Area = 352 sq units.

Problem 5: Triangle Side Relations

Given: Triangle sides where C > B > A, with conditions 2A + 7C = 9B, A = 12 Goal: Find length of C.

Steps:

  1. Recognize ABC must form a triplet (8, 15, 17), scaled appropriately.
  2. Use triplet recognition: 1.5 times scale.
  3. Answer: C (hypotenuse) = 25.5.

3. Key Concepts and Formulas

  1. Inradius of Right-Angle Triangle: in-radius = semi-perimeter - hypotenuse
  2. Area Calculation: Using inradius and semi-perimeter
  3. Pythagorean Triplets: Memorizing common triplets and scaling for solutions
  4. Geometry Intuition: Recognizing patterns and using logical deductions

4. Conclusion

  • Focused on problem-solving techniques and in-depth understanding of geometric properties.
  • Highlighted practical applications of Pythagorean theorem and inradii calculations.

By Ravi Prakash, Six Plus or Minus Geometry Course