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Question

In the given figures, O is the centre of the circles. Find the respective values of x.

A
60°
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B
115°
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C
40°
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Solution

In the first image,we see that AB=CD.
We know that equal chords subtend equal angles at the centre.
AOC=BOD=60

In the second image,AB=CD and AOB=40.
Since equal chords subtend equal angles at the centre, we have
AOB=COD=40.
Note that AOB and COD are isosceles.
Thus, using angle sum property in COD, we have
COD+OCD+ODC=180.
40+x+x=180
x=70

In the third image,
OCA+OCB=180°
OCB=105°
Now, using angle sum property in OCB, we have
30+105+BOC=180BOC=40.
Now, in ΔACO
OAC+OCA+AOC=180°
30°+75°+AOC=180°
AOC=75.
Thus, x=AOC+BOC=75°+40°=115°

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