011-40705070  or  
Call me
Download our Mobile App
Select Board & Class
  • Select Board
  • Select Class
Simranjeet Singh , asked a question
Subject: Math , asked on 1/3/13

the bisectors of the opposite angles A and C of cyclic quadrilateral ABCD intersect the circle at the point E and F.prove that EF is a diameter of the circle.

Ayushi Sharma From M.g.m. Sr Sec School, added an answer, on 1/3/13
332 helpful votes in Math

Given: ABCD is a cyclic quadrilateral. AE and CF are the bisectors of ∠A and ∠C respectively.

To prove: EF is the diameter of the circle i.e. ∠EAF = 90°

Construction: Join AE and FD.

Proof:

ABCD is a cyclic quadrilateral.

∴ ∠A + ∠C = 180° (Sum of opposite angles of a cyclic quadrilateral is 180° )

⇒ ∠EAD + ∠DCF = 90° ...(1) (AE and CF are the bisector of ∠A and ∠C respectively)

∠DCF = ∠DAF ...(2) (Angles in the same segment are equal)

From (1) and (2), we have

∠EAD + ∠DAF = 90°

⇒ ∠EAF = 90°

⇒ ∠EAF is the angle in a semi-circle.

⇒ EF is the diameter of the circle.

  • Was this answer helpful?
  • 4
View More 1 Answer
Ayushi Sharma From M.g.m. Sr Sec School, added an answer, on 1/3/13
332 helpful votes in Math

Given: ABCD is a cyclic quadrilateral. AE and CF are the bisectors of ∠A and ∠C respectively.

To prove: EF is the diameter of the circle i.e. ∠EAF = 90o

Construction: Join AE and FD.

Proof:

ABCD is a cyclic quadrilateral.

∴ ∠A + ∠C = 180o (Sum of opposite angles of a cyclic quadrilateral is 180o )

⇒ ∠EAD + ∠DCF = 90o ...(1) (AE and CF are the bisector of ∠A and ∠C respectively)

∠DCF = ∠DAF ...(2) (Angles in the same segment are equal)

From (1) and (2), we have

∠EAD + ∠DAF = 90o

⇒ ∠EAF = 90o

⇒ ∠EAF is the angle in a semi-circle.

⇒ EF is the diameter of the circle.

  • Was this answer helpful?
  • 1

Add an Answer

Start a Conversation
You don't have any friends yet, please add some friends to start chatting
Unable to connect to the internet. Reconnect
friends:
{{ item_friends['first_name']}} {{ item_friends['last_name']}}
{{ item_friends['first_name']}} {{ item_friends['last_name']}}
{{ item_friends["first_name"]}} {{ item_friends["last_name"]}} {{ item_friends["subText"] }}
{{ item_friends["notification_counter"]}} 99+
Pending Requests:
{{ item_friends['first_name']}} {{ item_friends['last_name']}}
{{ item_friends['first_name']}} {{ item_friends['last_name']}}
{{ item_friends["first_name"]}} {{ item_friends["last_name"]}} {{ item_friends["school_name"] }}
Suggested Friends:
{{ item_friends['first_name']}} {{ item_friends['last_name']}}
{{ item_friends["first_name"]}} {{ item_friends["last_name"]}} {{ item_friends["school_name"] }}
Friends
{{ item_friend["first_name"]}} {{ item_friend["last_name"]}} {{ item_friend["school_name"] }}
Classmate
{{ item_classmate["first_name"]}} {{ item_classmate["last_name"]}} {{ item_classmate["school_name"] }}
School
{{ item_school["first_name"]}} {{ item_school["last_name"]}} {{ item_school["school_name"] }}
Others
{{ item_others["first_name"]}} {{ item_others["last_name"]}} {{ item_others["school_name"] }}