Revision: Circle Geometry Maths 2 SSC (English Medium) 10th Standard Maharashtra State Board

A secant is a straight line that passes through the circle and intersects it at two distinct points. 

line AB cuts the circle at points M and N ⇒ AB is a secant.

A tangent is a straight line that touches a circle at exactly one point only, without cutting through it. This single point where the tangent touches the circle is called the point of contact or point of tangency.

Line CD touches the circle at point P only ⇒ CD is a tangent

Point P is the point of contact.

Central Angle: An angle whose vertex is the centre of the circle is called a central angle.

Two circles are touching if they intersect at exactly one point.

Types:

• Externally touching circles
  Distance between centres = sum of radii

• Internally touching circles
  Distance between centres = difference of radii

An inscribed angle is an angle whose vertex lies on the circle and whose arms intersect the circle at two other distinct points.

The arc of the circle intercepted by the arms of the angle is called the intercepted arc of the inscribed angle.

Given: AB and CD are tangents to both circles.

To prove: seg AB ≅ seg CD

Construction: Extend seg AB and seg CD to intersect each other at point E, such that A – B – E, C – D – E.

Proof: In above figure,

`{:(AE = CE),(BE = DE):}}`   ...(i) [Tangent segment theorem]

Consider, AE = CE

∴ AB + BE = CD + DE   ...[A – B – E, C – D – E]

∴ AB + DE = CD + DE   ...[From (i)]

∴ AB = CD

∴ seg AB ≅ seg CD

Given: A is the center of the circle. Tangents through external point D touch the circle at the points P and Q.

To prove: seg DP ≅ seg DQ

Construction: Draw seg AP, seg AD, and seg AQ.

Proof: In ΔPAD and ΔQAD,

seg PA ≅ seg QA    ...[Radii of the same circle]

seg AD ≅ seg AD    ...[Common side]

∠APD = ∠AQD = 90°    ...[Tangent theorem]

∴ ΔPAD ≅ ΔQAD    ...[By Hypotenuse side test]

∴ seg DP ≅ seg DQ  ...[Corresponding sides of congruent triangles]

Chord AB ≅ Chord CD   ...(Given)

∴ Arc ACB ≅ Arc CBD   ...(Arcs corresponding to congruent chords.)

∴ m(arc ACB) = m(arc CBD)   ...(1)

But m(arc ACB) = m(arc AC) + m(arc CB)   ...(2)

and m(arc CBD) = m(arc CB) + m(arc BD)   ...(3)

From (1), (2) and (3), we get,

m(arc AC) + m(arc CB) = m(arc CB) + m(arc BD)

∴ m(arc AC) = m(arc BD)

∴ arc AC ≅ arc BD

Hence, proved.

Chords Intersecting Inside:

AE × EB = CE × ED

Chords Intersecting Outside:

AE × EB = CE × ED

Tangent–secant segments:

EA × EB = ET²

Statement:
The measure of an inscribed angle is half of the measure of the arc intercepted by it.

Measure of an inscribed angle = `1/2` × measure of intercepted arc

• Angles inscribed in the same arc are congruent.
• An angle inscribed in a semicircle is a right angle.

1. Arc Definition: An arc is a curved portion of a circle's circumference between two points.

2. Two Types: Minor arc (< 180°) and Major arc (> 180°).

3. Semicircle: When the arc angle is exactly 180°, it's called a semicircle.

4. Complete Circle: Minor arc + Major arc = 360° (complete circumference).

A cyclic quadrilateral: All four vertices lie on the same circle

Key properties:

• Opposite angles are supplementary

• Exterior angle = interior opposite angle

Two arcs are congruent if:

• They belong to the same or congruent circles

• They have equal radii

• They have equal measures

Congruent arcs ⇔ congruent chords

• An angle whose vertex is the centre of a circle is called a central angle.

• Measure of minor arc = measure of its central angle

• Measure of major arc = 360° − minor arc

• The measure of a semicircle is 180°

• Full circle = 360°

1) Chords intersect inside the circle

Angle = `1/2` (sum of intercepted arcs)

(2) Secants intersect outside the circle

Angle = `1/2`(difference of intercepted arcs)

(3) Tangent–secant angle

Angle = `1/2` (intercepted arc)

These concepts unify angles + arcs.