Given - AB and CD are chords of a circle which intersect at right angles.
To Prove - arc CXA + arc DZB = arc AYD + arc BWC
Construction -
Join OA , OC , OD, OB , AD and CB
Let angle BCD be x.
Therefore angle DOB = 2x ( angle subtended by a segment at center is double the angle subtended by it at circumference; In this case DB subtends x on circumference so it'll subtend 2x at center)
Since angle BCD = x angle ABC = 180-(90+x) = 90-x ( angle sum property of triangle)
angle AOC = 2(angle ABC) [ angle subtended at the center is double the angle subtended at arc) → angle AOC = 2(90-x) = 180-2x
angle AOC + angle DOB = angle AOD + angle COB → arc CXA + arc DZB = arc AYZ + arc BWC [ since sum of the angles subtended by the arcs is equal , length of arcs will also be equal]
Lipimishra2
@siddhartha. I have a doubt. I read your ans for the first time now. The triangle AOD. O is the centre. Not the point where the 2 chords meetm
Lipimishra2
I guess you assumed that point as O, didn't you?
Answers & Comments
Verified answer
Given -AB and CD are chords of a circle which intersect at right angles.
To Prove - arc CXA + arc DZB = arc AYD + arc BWC
Construction -
Join OA , OC , OD, OB , AD and CB
Let angle BCD be x.
Therefore angle DOB = 2x ( angle subtended by a segment at center is double the angle subtended by it at circumference; In this case DB subtends x on circumference so it'll subtend 2x at center)
Since angle BCD = x
angle ABC = 180-(90+x) = 90-x ( angle sum property of triangle)
angle AOC = 2(angle ABC) [ angle subtended at the center is double the angle subtended at arc)
→ angle AOC = 2(90-x) = 180-2x
Therefore, angle AOC + angle DOB = 180-2x + 2x = 180°
→ angle AOC + angle DOB = 180°
Hence , angle AOD + angle COB = 360 - (angle AOC + angle DOB)
→ angle AOD + angle COB = 360 - 180° = 180°
Now since :
angle AOC + angle DOB = angle AOD + angle COB
→ arc CXA + arc DZB = arc AYZ + arc BWC
[ since sum of the angles subtended by the arcs is equal , length of arcs will also be equal]
Verified answer
Given that 2 chords AB and CD of a circle intersect at right angles.In triangle AOD:
We know that Sum of all angles of triangle = 180.
DOA + ODA + AOD = 180
DOA + ODA + 90 = 180
DOA + ODA = 180 - 90
DOA + ODA = 90 ---- (1)
Now, angle DOA & ODA are subtended by arc DZB & arc CXA respectively.
arc DZB + arc CXA = 90 ------ (2).
In triangle DOB:
BOD + OBD + DOB = 180
BDO + OBD + 90 = 180
BDO + OBD = 90 - ------ (3)
Now, angle BDO and OBD are subtended by arc BWC & arc AYD respectively.
arc BWC + arc AYD = 90 ------- (4)
From (3) & (4),
arc DZB + arc CXA = arc BWC + arc AYD = 90.
arc DZB + arc CXA = arc BWC + arc AYD = semicircle.
Note: Sorry, I didn't solve it on my own.
Hope this helps!