Geometry Class 10: Finding the Number of Triangles from Given Perimeter

Introduction

• Topic: Finding the number of triangles based on the given perimeter.
• Main takeaway: There are formulas you can use to quickly determine the number of distinct triangles.

Key Points

Formulas Based on Perimeter

• If the perimeter (P) is even:

  • Number of distinct triangles = (\left\lfloor \frac{P^2}{48} \right\rfloor)

• If the perimeter (P) is odd:

  • Number of distinct triangles = (\left\lfloor \frac{(P + 3)^2}{48} \right\rfloor)

Nearest Integer Function

• The expression (\left\lfloor x \right\rfloor) refers to the nearest integer function.
• Rules for nearest integer:
  • If the value is 3.8, the result is 4.
  • If the value is 3.4, the result is 3.
  • If the value is 3.5, the result is 3 (if you follow the rule to round down).

Examples

Example 1: Perimeter = 16 (Even)

• Calculation: (\left\lfloor \frac{16^2}{48} \right\rfloor = \left\lfloor \frac{256}{48} \right\rfloor)
• Results: (256 < 288), hence always less than 5.5 so nearly 5.4 -> Answer is 5 distinct triangles.

Example 2: Perimeter = 27 (Odd)

• Calculation: (\left\lfloor \frac{(27 + 3)^2}{48} \right\rfloor = \left\lfloor \frac{30^2}{48} \right\rfloor)
• Results: Answer is 19 distinct triangles.

Scalene Triangles

• Use similar formulas, with adjusted P values.
• For odd perimeter: (\left\lfloor \frac{(P - 3)^2}{48} \right\rfloor)
• For even perimeter: (\left\lfloor \frac{(P - 6)^2}{48} \right\rfloor)

Example: Perimeter is 24

• Calculation for scalene: (\left\lfloor \frac{(24 - 6)^2}{48} \right\rfloor) gives 7 scalene triangles.

Finding Isosceles Triangles

• No direct formula; use relationship:
  • Total triangles = Scalene + Isosceles + Equilateral.
• Example formula for Isosceles:
  • Total triangles - Scalene triangles - Equilateral triangles = Isosceles triangles

Example: Perimeter = 30

• Find Total: 19, Scalene: 12, Equilateral: 1 -> Isosceles = 6.

More Advanced Questions

• Given perimeter = 80
• Solve based on conditions:
  • Case 1: All sides even.
  • Case 2: Two sides odd, one side even.

Calculations

1. For all sides even: (\left\lfloor \frac{40^2}{48} \right\rfloor = 33.**
2. For two sides odd, one side even: total from even subtracting above gives 100 distinct triangles.

Summary

• Understanding the formulas for perimeter related triangle calculations can save time and is essential for geometry problems in exams.