Theorems on Chord and Subtended Angle

Here we will discuss some theorems on chord and subtended angle.

In this Circle -Theorems, the theorems are based on equal
chord. Here, we shall also study theorems on subtended angles and the relationship between the lengths of chords and their distances from the center of the given circle.

Theorems on Chord and Subtended Angle

1) Equal chords of a circle are equidistant from the center.
Given : Two chords AB and CD of circle C(O,r) such that
AB = CD and OL ⊥ CD.
Prove that : Chords AB and CD are equidistant from the center
i.e. OM = OL
Construction : Join OA and OC.

1) OL ⊥ AB ⇒ AL = 1/2AB 1) OM ⊥ CD ⇒ CM = 1/2CD
2) AB = CD 2) Given
3) 1/2 AB = 1/2 CD 3) Given
4) AL = CM 4) From AB = CD
5) OA = OC 5) Equal radius of the circle
6) AL = CM 6) From (4 )
7) ∠ALO = ∠CMO 7) Each 900
8) OL = OM 8) CPCTC
2) Equal chords of a circle subtend equal angles at the center.
Given : AB = CD
Prove that : ∠AOB = ∠COD

Statements
Reasons
1) AB = CD 1) Given
2) OA = OC 2) Radius of same circle
3) OB = OD 3) Radius of same circle
4) ΔAOB = ΔCOD 4) SSS Postulate
5) ∠AOB = ∠COD 5) CPCTC

3) Angles in the same segment of a circle are equal.
Given: A circle C(O,r), an arc PQ and two angles ∠PRQ and ∠PSQ
in the same segment of the circle.
Prove: ∠PRQ = ∠PSQ
Construction: Join OP and OQ.

Statements
Reasons
1) ∠POQ = 2∠PRQ 1) Given
2) ∠POQ = 2∠PSQ 2) Given
3) 2∠PRQ = 2∠PSQ 3) From (1) and (2)
4) ∴ ∠PRQ = ∠PSQ 4) From (3)
5) Reflex ∠POQ = 2∠PRQ and Reflex ∠POQ = 2∠PSQ 5) From the diagram
6) 2∠PRQ = 2∠PSQ 6) From the diagram
7) ∠PRQ = ∠PSQ 7) From (6)


Theorems on Circle

Theorems on Chord
Theorems on Chord and Subtended Angle
Theorems on Arc and Angle
Theorems on Cyclic Quadrilateral

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