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Geometry - Angles in a semi-circle+triangle

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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Sat Mar 21, 2015 11:00 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 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:
Image

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

So, ∆ABO is an isosceles triangle.
Image

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:
Image

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

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

Finally, we can see that the 3 angles with blue circles around them are on a line.
Image
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:
Image

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

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

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

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

Answer = D

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Brent
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by binaras » Sat Mar 21, 2015 11:07 am
Thanks for the quick response. Appreciate it.

Binara
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by GMATGuruNY » Sat Mar 21, 2015 12:17 pm
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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by jain2016 » Tue May 03, 2016 9:37 am
Since angles on a LINE must add to 180º, we know that ∠OBC = 2x
Image

Hi Brent ,

Can you please advise how come ∠OBC = 2x ?

Please explain.

Many thanks in advance.

SJ
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by [email protected] » Tue May 03, 2016 9:53 am
Hi jain2016,

That conclusion is based on two geometry rules:

1) The 3 angles in a triangle sum to 180 degrees.
2) The angles on a line sum to 180 degrees.

Once you know that Triangle ABO is ISOSCELES, and those two angles are both X, the 3rd angle in that triangle will equal (180-2X).

Since... X + X + (180-2X) = 180 degrees

Since that angle of the line ABC is (180-2X), the OTHER angle will be 2X.

Since (180-2X) + 2X = 180 degrees

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