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# In the figure attached: ABC is a right triangle touching a

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Director
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In the figure attached: ABC is a right triangle touching a [#permalink]

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21 Dec 2006, 06:25
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In the figure attached: [note: figure not drawn to scale]

ABC is a right triangle touching a semicircle with center O. If angle BOD = 60 degrees and segment AC = 12. What is the length of the curve segment on the semicircle between points D and E ?

(1) angles BOD = COE

(2) angles BAC = BCA

Tell me how you like this question. I actually made it up !
Attachments

geometry_1.jpg [ 9.41 KiB | Viewed 542 times ]

Last edited by Mishari on 21 Dec 2006, 16:22, edited 3 times in total.
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21 Dec 2006, 06:49
Going for C

From 1) Only angles are given ...no info on radius ...can't calculate

From 2) can calculate the radius, but no info on angle inside the semicircle..can't calculate

Together -
Can calculate radius and then the arc..

BOE + COD = 2 BOE = 120

So angle EOD = 60
... so the arc (2 pi r) * (60 / 360)

but the diagram is not as per the information given..it will look good if we swap D and E...
Generally, these types of questions come with a statutory warning ... not as per the scale ...

By the way, the question will be good if it says BOE = COD = 120, answer will be same
Director
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21 Dec 2006, 06:54
THanks alot anindyat

I modified the question. I guess now it looks alot better
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21 Dec 2006, 09:13
I have a question regarding the question.

If BOE=60 then BOD can't be equal to COE.

If that BOE is a typo for DOE then we need only the radius so B is the ans

Guess you need to correct the question again.

P.S.: Good thinking, good question!
Director
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21 Dec 2006, 16:24
THanks Sumithra

I wrote the question first on a piece of paper. When i typed the question of my computer, i kinda missed up some of the angles

The problem is fixed right now and everything makes a perfect sense.
Director
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22 Dec 2006, 01:43
OA is C

Here is why:

Given: BOD = 60 degrees and AC = 12

(1) BOD = COE
-------------------
if BOD = COE = 60 degrees, then we can calculate the value of the angle DOE = 180 - (2 x 60 ) = 60

statement 1 is insufficient

(2) BAC = BCA
--------------------
Here we know that the right triangle is actually an isosceles triangle
therefore, AB = BC and angles BAC = BCA = 45 degrees

Therefore, we can find the length of BC = 12 / sqrt(2)
--> the radius of the semicircle = BC/2 = 6/sqrt(2)
The circumference of the circle is 2 pi * 6/sqrt(2) = 12 pi/ sqrt(2)

yet, we know nothing about the angle DOE

statement 2 is insufficient

(1) and (2) together
-------------------------
from a property of circles:

curve DE/ circumference of the circle = angle DOE / 360

DE / ( 2 pi / sqrt(2) ) = 60/360

--> curve DE = sqrt (2) / 12 pi

22 Dec 2006, 01:43
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