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Difficulty: 505-555 Level,   Geometry,                     
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In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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Sol- we are given and also by second theorem that z^2=xy
1- this gives y=2x--> z^2=2x^2
2-\(z=5\sqrt{2}\)...not given x or y

Combining we know
\(5\sqrt{2}^2=2x^2\)
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
From 1 statement we know y = 2x
And then z = x√3 but to calculate z we need it's value
From 2 statement we know will get z
So it's together sufficient

study mode
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
carcass , Can you please let us know . How this question can be solved.

Thanks.
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
Given: z = Root(X*Y) -> to answer question we need to have information about z and y

1) y = 2x -> z=Root(x*2x) = Root (2x^2) = x*root(2) -> however INSUFF. as we have no information about x
2) z=5*root(2) clearly Insuff as we don't have any details about x or y

However combining (1) + (2) you get x*root(2) = 5*root(2) => x=5 , therefor SUFFICIENT
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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prashant0099 wrote:
carcass , Can you please let us know . How this question can be solved.

Thanks.


Hi

You can go through the solutions posted in this thread.
You can also read about circles theory here:

https://gmatclub.com/forum/math-circles-87957.html
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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Bunuel, chetan2u could you please share your approach here? Thanks.
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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In the OG2018 explanation of this answer it is said that only by looking at statement 2) one can conclude that CD is a diameter of the circle. How do we get to that conclusion?
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In the figure above, AB, which has length z cm, is tangent to the [#permalink]
amanvermagmat wrote:
prashant0099 wrote:
carcass , Can you please let us know . How this question can be solved.

Thanks.


Hi

You can go through the solutions posted in this thread.
You can also read about circles theory here:

https://gmatclub.com/forum/math-circles-87957.html



hey generis today is international day of indepencence :) so i tried to solve the above DS question independently, :grin: i ve read whole thread also visited this link...

but i smply couldnt understand how to approach the problem...

i know formula of area of circle PiR^2 and also circumference 2PiR

i know what is tangent, what is diametr and radius, but still couldnt wrap my mind aroud this problem what rule to apply here , can you help please :-)

have a fantastic day :)
Attachments

circle1.jpg
circle1.jpg [ 13.96 KiB | Viewed 30382 times ]

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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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generis many thanks for great explanation... :) just in the end the last part about math it was like "tail wagging the dog " :lol: :-)
you know i think the root of my gmat quant problem, is in root :lol:

If you need to, do the math. (i would love to do the math) :)

\((5\sqrt{2})^2=2x^2\) why after you sqauare both sides

\(50=2x^2\) --> the 2x^2 remains intact? :?

OR why didnt you write initially

\((5\sqrt{2})^2=(2x^2)^2\) you squared RHS, why didnt you apply the same pattern to \(2x^2\) :?
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In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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dave13 wrote:
generis many thanks for great explanation... :) just in the end the last part about math it was like "tail wagging the dog " :lol: :-)
you know i think the root of my gmat quant problem, is in root :lol:

If you need to, do the math. (i would love to do the math) :)

\((5\sqrt{2})^2=2x^2\) why after you sqauare both sides

\(50=2x^2\) --> the 2x^2 remains intact? :?

OR why didnt you write initially

\((5\sqrt{2})^2=(2x^2)^2\) you squared RHS, why didnt you apply the same pattern to \(2x^2\) :?

dave13 - too much staring at numbers? :)

I would bet you missed the difference between \(z^2\) and \(z\)

The equation is: \(z^2=2x^2\)

\(z\) by itself is: \(z=5\sqrt{2}\) -- Not squared yet. Plug \(z\) INTO the \(z^2\) equation

\(z^2=2x^2\)
\((z)*(z) =2x^2\) Leave RHS alone. That's what \(z\)-squared equals :)
\(z=5\sqrt{2}\)
\((z)*(z) =2x^2\)
\((5\sqrt{2})*(5\sqrt{2})=2x^2\)
\(5*5*\sqrt{2}*\sqrt{2}=2x^2\)
\(25*2=2x^2\)
\(50=2x^2\)
\(\frac{50}{2}=x^2\)
\(25=x^2\)
. Take the square root of both sides:

\(x=5\) (lengths cannot be negative)

I hope that helps. :-)
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
Hi generis,

Thanks for your explanation. Could you copy here your explanation to mikedays' query?

Thanks!!
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
Would someone kindly look at the diagram from the solution given by OG?

Based on ONLY the 2nd data point (2): z = 5√2

How are we able to determine that CD is the diameter of the circle?
And that the triangle AOB is a right-angle triangle?
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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Hi lassakul

Sometimes the explanations in the OG are just complex and not easy to follow or comprehend.

I would advise you to look at it from the power of point theorem as explained in the above posts.
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
In the figure above, AB, which has length z cm, is tangent to the circle at point A, and BD, which has length y cm, intersects the circle at point C. If BC = x cm and \(z = \sqrt{xy}\), what is the value of x ?

(1) CD = x cm

This mean y = 2x

INSUFFICIENT.

(2) \(z = 5 \sqrt{2}\)

INSUFFICIENT.

(1&2)
\(z = \sqrt{xy} = 5 \sqrt{2}\)

\(\sqrt{xy} = \sqrt{2x^2} = \sqrt{2}x\)

\(x = 5\)

SUFFICIENT.

Answer is C.
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
In the figure above, AB, which has length z cm, is tangent to the circle at point A, and BD, which has length y cm, intersects the circle at point C. If BC = x cm and z=√xy, what is the value of x ?

(1) CD = x cm
This tells us : z=√2x^2
Insufficient.

(2) z=5√2
Clearly insufficient

C: Sufficient.
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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carcass wrote:

In the figure above, AB, which has length z cm, is tangent to the circle at point A, and BD, which has length y cm, intersects the circle at point C. If BC = x cm and \(z = \sqrt{xy}\), what is the value of x ?

(1) CD = x cm

(2) \(z = 5 \sqrt{2}\)


Attachment:
circle.jpg

Solution:

We need to determine the value of x.

Statement One Alone:

Notice that the combined length of CD and BC equals the length of BD; thus, y = 2x. Substituting this into the equation z = √(xy), we obtain:

z = √(xy) = √(x * 2x) = √(2x^2) = x√2

Since we know nothing about the value of z, we cannot determine the value of x. Statement one alone is not sufficient.

Statement Two Alone:

We know z = 5√2 and we know z = √(xy); however, without a relation between x and y, this is not enough information to calculate the value of x. For instance, if y = 25 and x = 2; then √(xy) = √(50) = 5√2. In this case, the value of x is 2. On the other hand, if y = 10 and x = 5, then again √(xy) = 5√2, but in this case, the value of x is 5.

Statement two alone is not sufficient.

Statements One and Two Together:

Using the analysis of statement one, we know z = x√2 and using statement two, we know z = 5√2. Setting the two expressions of z equal to each other, we obtain x√2 = 5√2, which implies x = 2. Statements one and two together are sufficient.

Answer: C
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Re: In the figure above, AB, which has length z cm, is tangent to the [#permalink]
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