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If E is a point on the circumference of the circle circumscribing squa

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If E is a point on the circumference of the circle circumscribing squa [#permalink]

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New post 11 Jul 2016, 14:39
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A
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C
D
E

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Question Stats:

40% (01:12) correct 60% (02:10) wrong based on 52 sessions

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If E is a point on the circumference of the circle circumscribing square ABCD, as shown above, what is the value of x?

A. 45
B. 90
C. 120
D. 135
E. 270
[Reveal] Spoiler: OA

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Re: If E is a point on the circumference of the circle circumscribing squa [#permalink]

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New post 11 Jul 2016, 21:55
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Method 1:

The square's vertices divide the circle's circumference (360°) into 4 equal arcs of 90° each.
The angle x is an inscribed angle that defines an arc that is made of three 90° arcs (BC, CD, DA).
Hence, x defines a 3×90° arc.

Remember: The arc is twice the inscribed angle that defines it

So, the angle x is half the 3×90° arc --> x = 3×45° = 135°

Answer D.

Method 2:

We know that ∠DEB = 90 (angle subtended at the circumference by the diameter is always 90º).
So, ∠AEB is surely > 90. Option A and B out.

∠AEB will surely be <180. (obvious reasons) Option E is out.

We are now left with 120 and 135. Now, draw the same figure on paper to guess that the angle will be mostly greater than 120.
I did the same thing and guessed D in 30 secs.

Hope you find this helpful.

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If E is a point on the circumference of the circle circumscribing squa [#permalink]

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New post 11 Jul 2016, 22:57
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On drawing lines from the centre of the circle (assume it to be O),
we get (diagonals of a square are perpendicular to each other)

now if again we draw two lines from A and B to the circumference of the circle , lets name it F
we get (since angle at the center is twice the angle at the circumference)
hence
again aebf is a cyclic quadrilateral,
so or
Note: if we can visualize all these, this would hardly take 30 secs
and apologies for the deformed figure :)

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Re: If E is a point on the circumference of the circle circumscribing squa [#permalink]

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New post 15 Nov 2017, 01:26
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Re: If E is a point on the circumference of the circle circumscribing squa   [#permalink] 15 Nov 2017, 01:26
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