Given that I have 3 equations and 5 unknowns
I need to know 2 unknowns so C ?
Inscribed triangle in a semicircle
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pepeprepa
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Thanks to angles equations we can find that the two small triangles are similar and then we can make ratios between the different sides of these triangles.
Then, we have 5 equations and 5 unknowns. Only one value is necessary to answer the question.
I would say D
Then, we have 5 equations and 5 unknowns. Only one value is necessary to answer the question.
I would say D
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santa_dem
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You can make ratios, but you do not know the ratio. There is know way to solve this only with one statement. You need to know at least 2 angles and 1 side to find the other sides. with only one side you have no chance.pepeprepa wrote:Thanks to angles equations we can find that the two small triangles are similar and then we can make ratios between the different sides of these triangles.
Then, we have 5 equations and 5 unknowns. Only one value is necessary to answer the question.
I would say D
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santa_dem
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I was refering to the value of the ratios.pepeprepa wrote:With the angles and the position of each side you can make ratios but you do not know the ratios values of course.
We will see after OA.
OA is D
I am unable to figure out how is it D
Let's consider the below diagram ,where in each of the angles is marked .
(The below figure BD is marked as 5 ,in the explanation of solution.Not sure how did he arrive )
w+x = 90 (1)
y +x = 90 (2)
Hence w=y (3)
Similarly ,y+z = 90 (4)
y+x = 90 (5)
Hence z=x (6)
Angle ADB = BDC = 90
Hence triangles are similar triangles and corrresponding sides are proportional
AD/BD = BD/DC = AB/BC
We know DC = 25 / 4 .But we do not know BD (not sure how the solution guy assumes its value as 5)
How is the OA D?
I am unable to figure out how is it D
Let's consider the below diagram ,where in each of the angles is marked .
(The below figure BD is marked as 5 ,in the explanation of solution.Not sure how did he arrive )
w+x = 90 (1)
y +x = 90 (2)
Hence w=y (3)
Similarly ,y+z = 90 (4)
y+x = 90 (5)
Hence z=x (6)
Angle ADB = BDC = 90
Hence triangles are similar triangles and corrresponding sides are proportional
AD/BD = BD/DC = AB/BC
We know DC = 25 / 4 .But we do not know BD (not sure how the solution guy assumes its value as 5)
How is the OA D?
Last edited by gmatblr on Wed Aug 13, 2008 8:31 am, edited 1 time in total.
pepeprepa ,
Can u just briefly say how did u so quickly deduce that there are 5 unknown and 5 equations .
Which 5 equations do u mean
And how about the 5 unknown ?
2 angles of 90 degree known
1 side is given ineach of the condition.
So how 5 unknown
It wd be helpful to know how could one quicly interpret such figures
Thanks in Advance
Can u just briefly say how did u so quickly deduce that there are 5 unknown and 5 equations .
Which 5 equations do u mean
And how about the 5 unknown ?
2 angles of 90 degree known
1 side is given ineach of the condition.
So how 5 unknown
It wd be helpful to know how could one quicly interpret such figures
Thanks in Advance
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pepeprepa
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First of all, when you find C with such a question, if you know that is a difficult one, I think it is wise to pick D, that is what I would have done even if I find C at first. C seems too obvious, don't you think?
y+z=90
x+y=90
w+x=90
w+z=90
y=w
x=z
We agree on that. Therefore, the two small triangles are similar.
We can write:
b/c=c/d=a/e
Which is also
bd=c^2
be=ca
And also:
c^2 + b^2=a^2
c^2 + d^2=e^2
a^2 + e^2=(b+d)^2
Finally, we have a,b,c,d,e which are unknowns and 5 inequalities, so we only need one value to find the others.
The thing is not to solve it but as soon as you have your ratios and your pythagore inequalities I think you have to choose D and that's all.
y+z=90
x+y=90
w+x=90
w+z=90
y=w
x=z
We agree on that. Therefore, the two small triangles are similar.
We can write:
b/c=c/d=a/e
Which is also
bd=c^2
be=ca
And also:
c^2 + b^2=a^2
c^2 + d^2=e^2
a^2 + e^2=(b+d)^2
Finally, we have a,b,c,d,e which are unknowns and 5 inequalities, so we only need one value to find the others.
The thing is not to solve it but as soon as you have your ratios and your pythagore inequalities I think you have to choose D and that's all.
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Thanks pepeprepa .
One last thing ,
I still don't get the reasoning behind BD = 5 .Is there any reasoning ,why was is it so obviously put in the soln as ,BD =5 .
I do not find any way how one can easily deduce the value of BD ,other than to solve the tough 5 equations which you had mentioned earlier .
One last thing ,
I still don't get the reasoning behind BD = 5 .Is there any reasoning ,why was is it so obviously put in the soln as ,BD =5 .
I do not find any way how one can easily deduce the value of BD ,other than to solve the tough 5 equations which you had mentioned earlier .
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Ian Stewart
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Out of curiosity, who is the 'solution guy'? If the only information given is the information provided in the original post, the answer is certainly C and not D. If you take statement 2, for example, and AD=4, the diameter of the circle could be 5, or it could be 5,000,000- there's no restriction on how long DC could be. To see this, choose any circle with a large diameter AC, put the point D four units away from A on the diameter, and you have the picture given in the question. So CD could really be anything if you only know Statement 2. Of course the logic is the same for Statement 1. Clearly the two statements together are sufficient.gmatblr wrote: We know DC = 25 / 4 .But we do not know BD (not sure how the solution guy assumes its value as 5)
How is the OA D?
You can only solve five linear equations (and actually, those equations aren't linear, but that's not so important here) if the equations are independent. You have bd = c^2. Add the two equations:pepeprepa wrote:
We can write:
b/c=c/d=a/e
Which is also
bd=c^2
be=ca
And also:
c^2 + b^2=a^2
c^2 + d^2=e^2
a^2 + e^2=(b+d)^2
Finally, we have a,b,c,d,e which are unknowns and 5 inequalities, so we only need one value to find the others.
The thing is not to solve it but as soon as you have your ratios and your pythagore inequalities I think you have to choose D and that's all.
c^2 + b^2=a^2
c^2 + d^2=e^2
b^2 + 2c^2 + d^2 = a^2 + e^2
b^2 + 2bd + d^2 = a^2 + e^2 (using the fact that bd = c^2)
(b+d)^2 = a^2 + e^2
So the equation (b+d)^2 = a^2 + e^2 is not a 'new' relationship among a, b, d and e, so doesn't count as an independent equation: we can derive this relationship from the other equations.
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