0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

In Fig. 10.19 AB and CD are two chords of a circle intersecting each other at point E Prove that AEC=12(Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre)
426720_5343a1370efb49fcba67eb61096d669d.png

Solution
Verified by Toppr

AB and CD are two chords of a circle intersecting each other at point E.

We have to prove that AEC=12 (Angles subtended by an arc CXA at the centre + angle subtended by arc DYB at the centre.)

Join AC.BC and BD

Since, the angles subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. now arc CXA subtends AOC at the centre and ABC at the remaining pan of the circle, so

AOC=2ABC.........(1)

Similarly, BOD=2BCD...........(2)

Now, adding (1) and (2), we get

AOC+BOD=2(ABC+BCD)........(3)

Since exterior angle of a triangle is equal to the sum of interior opposite angles,
so in ΔCEB, we have

AEC+ABC+BCD..........(4)

From (3) and (4), we get

AOC+BOD=2AEC

or AEC=12(AOC+BOD)

Hence, AEC=12 (angles subtended by an arc CXA at the centre +angle substended by an arc DYB at the centre)

Was this answer helpful?
3
Similar Questions
Q1
In Fig. 10.19 AB and CD are two chords of a circle intersecting each other at point E Prove that AEC=12(Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre)
426720_5343a1370efb49fcba67eb61096d669d.png
View Solution
Q2

In the given figure, AB and CD are two chords of a circle, intersecting each other at a point E. Prove that

AEC=12(angle subtended by arc CXA. at the centre + angle subtended by arc DYB at the centre).

View Solution
Q3
Question 8
In the figure, AB and CD are two chords of a circle intersecting each other at point E. Prove that AEC=12 (angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).

View Solution
Q4
Two chords AB and CD intersect at P inside the circle. Prove that the sum of the angles subtended by the arcs AC and BD at the centre O is equal to twice the angle APC.
View Solution
Q5
If two chords AB and CD of a circle AYDZBWCX intersect at right angles prove that arc(CXA)+arc(DZB)=arc(AYD)+arc(BWC)=arc(semicircle)


426712_9bf07ee6865f4817bf9b9d161e41d24e.png
View Solution