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In the figure shown, point O is the center of the semicircle [#permalink]

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15 Dec 2009, 13:09

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

(1) The degree measure of angle COD is 60º. (2) The degree measure of angle BCO is 40º.

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

Write down everything you know from the stem:

\(BO=CO=radius=AB\) --> triangles BOC and ABO are isosceles. \(\angle BAO = \angle BOA\) and \(\angle BCO = \angle CBO\) \(\angle CBO = 2*\angle BAO\)

(1) The degree measure of angle COD is 60º: \(\angle BAO +\angle ACO = \angle COD = 60º\) degrees (Using exterior angle theorem) \(\angle ACO = \angle CBO = 2* \angle BAO\) \(So, \angle BAO + \angle ACO = 2* \angle BAO + \angle BOA = 3* \angle BAO = 60º\) \(\angle BAO = 20º\). SUFFICIENT

(2) The degree measure of angle BCD is 40º: \(\angle BCO=40º\) --> \(\angle BCO = \angle CBO=40º = 2*\angle BAO\) --> \(\angle BAO=20º\). SUFFICIENT

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

Write down everything you know from the stem:

\(BO=CO=radius=AB\) --> triangles BOC and ABO are isosceles. \(\angle BAO = \angle BOA\) and \(\angle BCO = \angle CBO\) \(\angle CBO = 2*\angle BAO\)

(1) The degree measure of angle COD is 60º: \(\angle BAO +\angle ACO = \angle COD = 60º\) degrees (Using exterior angle theorem) \(\angle ACO = \angle CBO = 2* \angle BAO\) \(So, \angle BAO + \angle ACO = 2* \angle BAO + \angle BOA = 3* \angle BAO = 60º\) \(\angle BAO = 20º\). SUFFICIENT

(2) The degree measure of angle BCO is 40º: \(\angle BCO=40º\) --> \(\angle BCO = \angle CBO=40º = 2*\angle BAO\) --> \(\angle BAO=20º\). SUFFICIENT

I still miss it. I saw the Circle Chapter from the Maths book as well. Why is

<CBO=2<BAO

First of all note that ABO is isosceles triangle. Why? Given that AB=OC, OC is radius, but OB is also radius, hence AB=OC=OB=r --> two sides in triangle ABO namely AB and BO are equal. Which means that angles BAO and BOA are also equal.

So we have <BAO=<BOA.

Next step: angle <CBO is exterior angle for triangle BAO. According to the exterior angle theorem: An exterior angle of a triangle is equal to the sum of the opposite interior angles. --> <CBO=<BAO+<BOA, as <BAO=<BOA --> <CBO=<BAO+<BAO=2<BAO.

Hope it's clear.

For more about the triangles check the link abot triangles below.
_________________

Let angle BAO=x since AB=BO we have angle BOA=x SInce Angle CBO is an exterior angle to BAo and BOA it is equal to the sum of their individual angles Angle CBO = x+x=2x

BO and CO are the two radii hence they subtend equal angles thus BCO = 2x and BOC = 180-4x We need x Statement 1 gives COD Since COD+BOC+AOB = 180 60+180-4x+x=180 We cans olve for x - sufficient

Statement 2 gives BCO = 2x = 40

we can calculate x hence sufficient.

Answer is D (Hope this is clear.)
_________________

In the figure shown, point O is the center of the semicircle [#permalink]

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19 Dec 2009, 08:28

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

(1) The degree measure of angle COD is 60º. (2) The degree measure of angle BCO is 40º.

Attachment:

exteriorAngle.GIF [ 14.09 KiB | Viewed 86006 times ]

Re: In the figure shown, point O is the center of the semicircle [#permalink]

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18 Dec 2009, 18:33

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Given OC = AB OC = OB since both are radii. Therefore OB = AB. Since angles opposite to equal sides are equal. Therefore, Angle OCB = Angle OBC and Angle OAB = Angle BOA

Statement 1 Angle COD = 60 degrees. Not sufficient. Since we cannot determine angle COB or angle OCB. Statement 2 Angle OCB = 40 degrees. Not sufficient Since we cannot determine angle COB

Both together helps us determine Angle COB and angle BOA = 180 - angle COD - angle COB Therefore, answer is C
_________________

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

(1) The degree measure of angle COD is 60º. (2) The degree measure of angle BCO is 40º.

Target question:What is the degree measure of ∠BAO?

Given: The length of line segment AB is equal to the length of line sement OC

Statement 1: The degree measure of angle COD is 60º So, we have the following:

Since the radii must have equal lengths, we can see that OB = OC

So, ∆ABO is an isosceles triangle.

If we let ∠BAO = x degrees, then we can use the facts that ∆ABO is isosceles and that angles must add to 180º to get the following:

Since angles on a LINE must add to 180º, we know that ∠OBC = 2x

Now, we can use the facts that ∆BCO is isosceles and that the angles must add to 180º to get the following:

Finally, we can see that the 3 angles with blue circles around them are on a line. So, they must add to 180 degrees. We get: x + (180-4x) + 60 = 180 Simplify: 240 - 3x = 180 Solve to get: x = 20 In other words, ∠BAO = 20º Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The degree measure of angle BCO is 40º So, we have the following:

Since the radii must have equal lengths, we can see that OB = OC

So, ∆BCO is an isosceles triangle, which means OBC is also 40º

Since angles on a line must add to 180 degrees, ∠ABO = 140º

Finally, since ∆ABO is an isosceles triangle, the other two angles must each be 20º As we can see, ∠BAO = 20º Since we can answer the target question with certainty, statement 2 is SUFFICIENT

What have I missed? I too marked this as C. GMAT Prep says its D, so fine, agreed. But here is my reasoning. The reason is that it is not mentioned in the question that ABC is one single line. (points A,B,C all lie on the same line).

Why do we need to consider it as one straight line? If we do not consider them on the straight line, we cannot apply Exterior angle sum rule. If we cannot apply, we will not be able to solve this without both stmt1 and stmt2.

Cheers!

Ian Stewart:

"In general, you should not trust the scale of GMAT diagrams, either in Problem Solving or Data Sufficiency. It used to be true that Problem Solving diagrams were drawn to scale unless mentioned otherwise, but I've seen recent questions where that is clearly not the case. So I'd only trust a diagram I'd drawn myself. ...

Here I'm referring only to the scale of diagrams; the relative lengths of line segments in a triangle, for example. ... You can accept the relative ordering of points and their relative locations as given (if the vertices of a pentagon are labeled ABCDE clockwise around the shape, then you can take it as given that AB, BC, CD, DE and EA are the edges of the pentagon; if a line is labeled with four points in A, B, C, D in sequence, you can take it as given that AC is longer than both AB and BC; if a point C is drawn inside a circle, unless the question tells you otherwise, you can assume that C is actually within the circle;if what appears to be a straight line is labeled with three points A, B, C, you can assume the line is actually straight, and that B is a point on the line-- the GMAT would never include as a trick the possibility that ABC actually form a 179 degree angle that is imperceptible to the eye, to give a few examples).

So don't trust the lengths of lines, but do trust the sequence of points on a line, or the location of points within or outside figures in a drawing. "

What have I missed? I too marked this as C. GMAT Prep says its D, so fine, agreed. But here is my reasoning. The reason is that it is not mentioned in the question that ABC is one single line. (points A,B,C all lie on the same line).

Why do we need to consider it as one straight line? If we do not consider them on the straight line, we cannot apply Exterior angle sum rule. If we cannot apply, we will not be able to solve this without both stmt1 and stmt2.

So, if what appears to be a straight line is labeled with three points A, B, C, you can assume the line is actually straight, and that these points are actually on the line in the order given on the diagram.
_________________