Unit 12: Circle (वृत्त) Theorems & Key Points
Introduction
Unit 12: Circle (वृत्त) is a critical part of the Geometry syllabus. This unit deals with theorems related to arcs, chords, central angles, inscribed angles, cyclic quadrilaterals, and tangents. Understanding these properties is essential for solving geometric proofs and numerical problems.
Key Notations:
- $O$: Center of Circle (वृत्तको केन्द्र)
- $r$: Radius (अर्धव्यास)
- $d$: Diameter (व्यास)
- $\angle$: Angle (कोण)
- $\triangle$: Triangle (त्रिभुज)
- $||^{gm}$: Parallelogram
1. Angles in a Circle (वृत्तका कोणहरू)
[Image of Circle Theorems]| Theorem Name | Statement (कथन) |
|---|---|
| Central Angle & Inscribed Angle (केन्द्रिय कोण र परिधि कोण) |
The central angle is double the inscribed angle standing on the same arc. एउटै चापमा आधारित केन्द्रिय कोण परिधि कोणको दोब्बर हुन्छ । $$\angle AOB = 2 \angle ACB$$
|
| Angles in the Same Segment (एउटै खण्डका कोणहरू) |
Inscribed angles standing on the same arc are equal. एउटै चापमा आधारित परिधि कोणहरू बराबर हुन्छन् । $$\angle ACB = \angle ADB$$
|
| Angle in Semicircle (अर्धवृत्तको कोण) |
The angle in a semicircle is a right angle ($90^\circ$). अर्धवृत्तमा बनेको परिधि कोण समकोण ($90^\circ$) हुन्छ । |
2. Cyclic Quadrilateral (चक्रीय चतुर्भुज)
Properties:
-
Opposite Angles: The sum of opposite angles of a cyclic quadrilateral is $180^\circ$ (Supplementary).
चक्रीय चतुर्भुजका सम्मुख कोणहरूको योगफल $180^\circ$ हुन्छ ।$$\angle A + \angle C = 180^\circ, \quad \angle B + \angle D = 180^\circ$$ -
Exterior Angle: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
चक्रीय चतुर्भुजको बाहिरी कोण त्यसको आसन्न भित्री कोणको सम्मुख कोणसँग बराबर हुन्छ ।
3. Tangents (स्पर्शरेखाहरू)
| Property | Description |
|---|---|
| Radius-Tangent (अर्धव्यास र स्पर्शरेखा) |
The tangent at any point of a circle is perpendicular to the radius through the point of contact. वृत्तको अर्धव्यास र स्पर्शरेखा स्पर्श बिन्दुमा लम्ब हुन्छन् ($90^\circ$ बनाउँछन्) । $$O \perp Tangent$$
|
| Tangents from External Point (बाहिरी बिन्दुबाट स्पर्शरेखाहरू) |
The lengths of tangents drawn from an external point to a circle are equal. वृत्तको बाहिरी बिन्दुबाट खिचिएका दुई स्पर्शरेखाहरूको लम्बाइ बराबर हुन्छ । $$PA = PB$$
|
| Alternate Segment (एकान्तर खण्ड) |
The angle between a tangent and a chord is equal to the angle in the alternate segment. स्पर्शरेखा र जीवा बिचको कोण एकान्तर खण्डमा बनेको कोणसँग बराबर हुन्छ । |
Full Chapter PDF Manual
For detailed proofs and exercise problems, access the complete PDF manual below.