Geometry Practice Class Notes

Instructor: Ari Prakash

Key Concepts Covered

1. Condition for an Obtuse Angle Triangle
   • Longest side square: Greater than the sum of squares of the other two sides.
   • Formula: If longest side = c, then: c² > a² + b²

Exercise 1: Distance Between Incenter and Circumcenter

1. Given:
   • Obtuse triangle with one side = 180 cm; one side 80 cm => Isosceles Triangle
   • Possible configurations: 80, 50, 50 and 80, 80, 20
2. Solution Approach:
   • Verify configurations against the obtuse triangle condition.
   • Valid Configuration: 80, 50, 50
3. Finding Incenter (r):
   • Area Calculation:
     • Using Pythagoras theorem: Height = 30 cm (from median properties)
     • Area = 1/2 x Base x Height = 1/2 x 80 x 30 = 1200 cm²
     • Using relation: Area = r x s (semi-perimeter: s)
     • s = 1/2 of perimeter = 90 cm
     • `r = 13.33 cm (from 1200/90)
4. Finding Circumcenter (R):
   • Using: R = (a x b x c) / (4 x Area)
     • Calculation details: 80 x 50 x 50 / (4 x 1200)
     • R = 41.66 cm
5. Distance between Incenter and Circumcenter:
   • Sum of respective perpendiculars from the vertices of the base & height.
   • Result: Distance = 25 cm

Exercise 2: Tangents to a Circle

1. Given:
   • OA, OB (tangents); AP = 192; OP = 102 (each value halved, forming isosceles component)
   • Find radius (r)
2. Solution Approach:
   • Form kite: Using properties: Sum of squares, medians, intersection characteristics.
   • Key Calculations: Pythagoras
     • 204² + Radius² = 192²
     • Formula application leads resultantly to 108.8 cm.
   • Properties and relationships iteratively validated.

Review - Solution Verification

1. Key Verification Steps:
   • Validating Pythagoras, simplifying stated ratios of provided values.
   • Similar triangle relationships applied to precise axiom principles.

Questions of the Day

1. Good question adherence: Calculations rooted in understanding fundamentals.
2. Revising critical bounding algebra builds deeper exercise grasp.

Additional Tips:

1. Always reaffirm calculated tangents and their respective bisector properties.
2. Geometry consolidates deeply in working backward validations.

Questions practiced will elucidate optimal concept comprehensions.