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August 3rd, 2013, 04:54 AM   #1
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properties of circle doubts

Hi, I have encountered some problems with the last two part b(i) and b(ii) of the qns. I have no idea How I should solve it. However I have attached the answer and question.

[attachment=0:kspksmmc]20130803_204632.jpg[/attachment:kspksmmc]
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August 3rd, 2013, 06:44 AM   #2
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Re: properties of circle doubts

Given three points, it is always possible to draw a circle passing through them (and this defined circle is unique). So we can draw 2 circles passing through E,B,F and E,D,F. But as the angles EBF and EDF are equal, these 2 circles are congruent (overlapped).
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August 3rd, 2013, 06:45 AM   #3
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Re: properties of circle doubts

a) i) 90 ii) 33 iii) 44 iv) 68

b) i) Angle ABD = 68, so angle DBE = 112. DE and BF are the altitudes of triangle AEF, intersecting at C, so the extension of AC to EF is an altitude of triangle AEF and angle DFE is 68. 112 + 68 = 180, so EBDF is cyclic. ii) Triangle BEF is right angled, so its circumcenter lies on the midpoint of BF, its circumcircle having diameter EF. Hence the center of circle EBDF lies on the midpoint of EF.
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August 4th, 2013, 02:36 AM   #4
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Re: properties of circle doubts

Thanks alot
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