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655-705 (Hard)|   Geometry|      
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NateTheGreat11
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In the …Figure above, O is the center of a semicircle, and p Updated on: 05 Sep 2013, 10:00
Originally posted by NateTheGreat11 on 05 Sep 2013, 09:49.
Last edited by Bunuel on 05 Sep 2013, 10:00, edited 2 times in total.
Edited the question.
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

55% (hard)

Question Stats:

66% (02:29) correct 34% (02:37) wrong based on 128 sessions

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Attachment:
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SemiCircle.jpg [ 12.87 KiB | Viewed 4228 times ]
In the …Figure above, O is the center of a semicircle, and points A,B, C, D, and E lie on the semicircle. If the degree measure of angle AOC is 85, what is the degree measure of angle AOB?

(1) The degree measure of angle BOD is 100 degrees.

(2) The degree measure of angle COE is 95 degrees.

Source: Jeff Sackmann Geometry Challenge #97

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OA is E, I think it is A. Attached is my reasoning.

Find angle AOB, say x
Given angle AOC is 85 degrees, then angle COD is 95 degrees ( 85+95 = 180)
Given that angle BOD is 100 degrees
Thus
<BOD = (<AOC - <AOB ) + ( <EOC - EOD )
100 = (85-x) + (95-y)
80 = x +y

Since this is a semi circle the two 'pies'; angle AOC and angle EOC are similar, thus
<AOC is to <AOB as <EOC is to <EOD
(85/x) = (95/y)

X = 17/19Y
Y = 19/17X
80 = x + y

Solve for X (<AOB)
(I am unsure of this final step involving similarity... Is this permitted/justified?)
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E
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In the …Figure above, O is the center of a semicircle, and p 05 Sep 2013, 10:07
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NateTheGreat11

In the …Figure above, O is the center of a semicircle, and points A,B, C, D, and E lie on the semicircle. If the degree measure of angle AOC is 85, what is the degree measure of angle AOB?

(1) The degree measure of angle BOD is 100 degrees.

(2) The degree measure of angle COE is 95 degrees.

Source: Jeff Sackmann Geometry Challenge #97

OA is E, I think it is A. Attached is my reasoning.

Find angle AOB, say x
Given angle AOC is 85 degrees, then angle COD is 95 degrees ( 85+95 = 180)
Given that angle BOD is 100 degrees
Thus
<BOD = (<AOC - <AOB ) + ( <EOC - EOD )
100 = (85-x) + (95-y)
80 = x +y

Since this is a semi circle the two 'pies'; angle AOC and angle EOC are similar, thus
<AOC is to <AOB as <EOC is to <EOD
(85/x) = (95/y)

X = 17/19Y
Y = 19/17X
80 = x + y

Solve for X (<AOB)
(I am unsure of this final step involving similarity... Is this permitted/justified?)
Show more

The red part is not correct. Check the diagram below:
Attachment:
Angles.png
Angles.png [ 55.01 KiB | Viewed 4014 times ]
The correct answer is E.
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In the …Figure above, O is the center of a semicircle, and p 12 May 2016, 10:33
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E it is ..Neither of the two statements helps. Neither alone nor together ...
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In the …Figure above, O is the center of a semicircle, and p 13 May 2016, 05:21
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If we consider degree AOB=x, degree BOC=y, degree COD=z and degree DOE=w, there are 4 variables (x,y,z and w) and 2 equations (x+y+z+w=180 and x+y=85) in the original condition. In order to match the number of equations and the number of variables, there are 2 equations. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer.
Using the condition 1) and the condition 2) at the same time, we get y+z=100 and z+w=95. However, z+w=95 already appears in x+y=85. Hence, in overall, we lack 1 equation, which means the correct answer is E.



l For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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In the …Figure above, O is the center of a semicircle, and p 12 Jul 2018, 10:48
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E You cannot determine the exact value.
Also i do not think that you can use similarity with the sectors of a circle, you will have to make some audacious assumptions to do so.
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In the …Figure above, O is the center of a semicircle, and p 11 Aug 2019, 18:55
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