Path Angle Problems in Geometry – Must Know Concepts
May 30, 2024 2025-05-30 0:55Path Angle Problems in Geometry – Must Know Concepts
Path Angle Problems in Geometry – Must Know Concepts
Amiya Kumar 🚶 Path Problems in Geometry (Triangles) – CAT/XAT Level Concepts
🔹 Concept of Path Problems in Triangle
When a person or object traverses equal paths along the sides of a triangle and returns to the starting point (or ends at a symmetric location), we apply the Path Angle Closure Formula:
\[ \text{Angle at vertex} = \frac{180^\circ}{n} \quad \text{where } n = \text{number of equal segments} \]
Here, \( AB = BC = CA \), so \( n = 3 \Rightarrow \angle A = 60^\circ \)
🔹 Concept Applied with 5 Equal Paths
Let’s break triangle \( ABC \) into 5 equal segments and apply:
\[ \angle A = \frac{180^\circ}{5} = 36^\circ \]
🔹 Advanced Case: 7 and 9 Equal Paths
If a triangle is divided into 7 or 9 equal segments (symmetric routes), then the angle at the origin becomes:
- \( \angle A = \frac{180^\circ}{7} \approx 25.71^\circ \)
- \( \angle A = \frac{180^\circ}{9} = 20^\circ \)
🧠 Applied Angle Deduction Using Path Concept
Given 5 equal paths in triangle \( ABC \), angle at \( A \) is \( 36^\circ \). What is \( x \) in the setup where:
\[ 5x = 180^\circ \Rightarrow x = 36^\circ \]
🔍 7-Segment Full Angle Allocation
In this triangle, we see angle divisions done using equal paths, and the total rotation follows:
\[ 7x = 180^\circ \Rightarrow x = 25.71^\circ \]
📏 Highest Level Path Geometry – 9 Segment Distribution
Now, triangle is broken into 9 equal segments. Each internal corner angle is \( x = 20^\circ \). Beautiful path symmetry!
🌟 Bonus Concept: Isosceles Triangle Angle Deduction
In triangle \( ABC \), if two isosceles triangles are formed on same base, angle at apex \( \theta \) can be found using:
\[ \angle D = 180^\circ - 2\theta \]
🎯 Level 3: Two Concentric Circles – Angle in Triangle
Angle at center \( \angle BOD \)?
\[ \angle BOD = 180^\circ - 2x = 180^\circ - 100^\circ = 80^\circ \]
CAT 2025 Offline + Hybrid
