Can someone explain how he/she is getting the angle DOA (O being the centre of the circle) to be 90 degrees?

AB and AD are not parallel. You just copied the explanation from the end of the book. I have now officially wasted an entire day on a stupid question. Thanks to this WRONG explanation. I drew the actual circle and the chords and this question is incorrect!

This is the OE.



Hope this helps

Regards

Isn't the arc length of CD = 45/360 * 16 pi = 2 pi?

Here, the chords AD and BC are parallel, angle DAC is given 45, therefore, angle BCA is also 45. Now we cant directly calculate the length of given segments, we have to apply indirect method, total circumference= seg. BXC + seg. CD + seg. DYA + seg. AB, i.e. 16pi= x+ 2x+ CD+ AB( i assumed seg BXC to be x). Now, seg CD and seg AB are equal.
Reason- mark the center of the circle, now angle of the seg AB will be twice of the angle subtended by its chord AB(45), therefore, the angle is 90. Same goes for the other one and since they have the same angle, their length of the segment will be 2*pi*r*(90/360)=16pi/4=4pi.
Therefore, 16pi=3x+4pi+4pi, this gives us x=8pi/3. Hence lenght of seg BXC is 8pi/3.

We see that since arc CD corresponds to a 45 degree inscribed angle (angle CAD), arc CD is twice the measure of angle CAD. The measure of arc CD is 90 degrees, thus its arc length is 90/360 = 1/4 of the circle. We also can conclude that angle BCA is also 45 degrees (since BC is parallel to AD) and hence arc BA is also 90/360 = 1/4 of the circle. Since arc AYD is twice the length of arc BXC, we can let arc BXC = x (where x represents the fraction the arc length of BXC as of the circumference of the circle) and thus arc AYD = 2x. We can create the equation:

1/4 + 1/4 + x + 2x = 1

3x = 1/2

x = 1/6

Thus, arc BXC is 1/6 of the circumference of the circle, and arc AYD is 1/3 of the circumference of the circle.

Since the radius is 8, the circumference is 16π and thus arc BXC is 1/6 x 16π = 16π/6 = 8π/3.

Answer: B

The two arcs making 45 degree angle make up half the circumference. the remaining half will be shared by two arcs AYD and BXC in 2:1 ratio. So half the circumference is 8pi. which divided by 3 gives 8pi/3 the length of arc AXC. Answer B.

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