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# In the circle above with center O, minor arc AC is equal in length to

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Math Expert
Joined: 02 Sep 2009
Posts: 55230
In the circle above with center O, minor arc AC is equal in length to  [#permalink]

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06 Jun 2017, 11:19
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00:00

Difficulty:

45% (medium)

Question Stats:

68% (02:33) correct 32% (01:49) wrong based on 60 sessions

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In the circle above with center O, minor arc AC is equal in length to minor arc CF and line segment BD is parallel to line segment AF. If angle CAD = angle DAE = angle EAF, what is the measure of angle BDA?

A. 15
B. 30
C. 45
D. 60
E. 75

Attachment:

Circle_O.png [ 13.63 KiB | Viewed 1767 times ]

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Joined: 24 Apr 2016
Posts: 331
Re: In the circle above with center O, minor arc AC is equal in length to  [#permalink]

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06 Jun 2017, 13:36
1
In this problem, we can use the property :

In a circle, if two minor arcs are equal in measure, then their corresponding chords are equal in measure

Now if we join C to E, we get the Right Angled Traingle ACF

Since the minor arcs AC and CF are equal in length, correspoding chords AC and CF are also equal.

Therefore Traingle ACF is an Isocelese Right Angled Triangle.

Angle ACF = 90, Angle CAF = Angle CFA = 45

angle CAD + angle DAE + angle EAF = Angle CAF = 45

As angle CAD = angle DAE = angle EAF, then angle CAD = angle DAE = angle EAF = 15

angle BDA = angle DAE + angle EAF = 30

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Re: In the circle above with center O, minor arc AC is equal in length to  [#permalink]

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02 Jul 2017, 08:29

the angle subtended by arc AC and arc CF are equal, as a result of which angle subtended by both the acrs at the center is 90 (180 in total as AF is the diameter).

we know that the angle subtended at the circle is half the angle subtended at the centre.
also, it is given that angle CAD = angle DAE = angle EAF.

so angle subtended at the cirecumference of the circle by arc CF is 45.

since BD is parallel to AF, angle BDA is equal to angle DAF.

angle DAF = 30 (2/3rd of angle CAF )

Re: In the circle above with center O, minor arc AC is equal in length to   [#permalink] 02 Jul 2017, 08:29
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