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geometry - SemiCircle

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geometry - SemiCircle

by Xbond » Mon Aug 03, 2009 10:48 pm
Hi there,

I woud like your input about this difficult geometry question. Someone can explain to me in simple way the concept. Sorry, I don't know the answer. Look at the attachment.


In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of the line segment AB is equal to the length of linesegment OC, what is the degree measure of angle BAO ?

(1) The degree measure of angle COD is 60°.
(2) The degree measure of angle BCD is 40°.
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by Matmasi » Wed Aug 26, 2009 2:14 am
statement 1) cod = 60

What we see is that triangle OAB and triangle OBC are both isosceles, so they have two sides and two angles equal. We know that they are isosceles because they both have two sides that are equal to the radius of the semicircle.

So we know that angles OAB and AOB are equal. Let's call them x. So, angle OBA = 180-2x
For the properties of the sum f the exterior angle we know that OBC is equal to 180- (180-2x) = 2x.
Now, we know that also OBC and OCB are equal, so they are both 2x.

From statement 1 we know that angle COA is 120 and so 180=120+x+2x so x = 20
in other words 180= the sum of the angles of the triangle OCA

From statement 2:
BCO = 40
BCO =2x
x = 20

So the answer is D.




Now
BAO + COA + ACO = 180 and we know COA = 120
thus BAO + ACO = 60

Moreover we know that BO=CO cause they are radius, so the triangle OBC is isosceles.
So we also deduct that angle OCA and OBC must be equal
so OCA = OBC

so, bao + cbo =60
cbo = 2bao
hence bao = 20 and suff.

statement 2) aco = 40
aco = cbo
cbo = 2bao
hence bao = 20 and suff.
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by mcdesty » Tue Jul 08, 2014 5:50 pm
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by GMATGuruNY » Wed Jul 09, 2014 3:02 am
Image

In the figure shown, point O is the center of the semicircle and points B, C, and D lie on the semicircle. If the length of line segment AB is equal to the length of line sement OC, what is the degree measure of angle BAO?

(1) The degree measure of angle COD is 60.
(2) The degree measure of angle BCO is 40.
It is given that AB=OC.
Since OC and OB are both radii, OC=OB.
Thus:
Image

EVALUATE THE EASIER STATEMENT FIRST.
Since statement 2 gives information about one of the equal angles, start with statement 2.

Statement 2: The degree measure of angle BCO is 40.
The result is the following combination of angles:
Image
Thus, angle BAO = 20.
SUFFICIENT.

Statement 1: The degree measure of angle COD is 60.
In the combination of angles yielded by statement 2, angle COD = 60.
Thus, statement 1 implies the same combination of angles as does statement 2.
SUFFICIENT.

The correct answer is D.
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