Theorems on Chord and Subtended Angle

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

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

Home Page

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.