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M21-20

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M21-20 [#permalink]

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If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\)
B. \(\sqrt{20}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)
[Reveal] Spoiler: OA

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Re M21-20 [#permalink]

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New post 16 Sep 2014, 00:11
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Official Solution:

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\)
B. \(\sqrt{20}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)


Look at the diagram below:

Image

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).


Answer: B
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New post 12 Oct 2014, 19:55
I am having difficulties with this problem I actually got it right but I might have been lucky I tried to find the slope which i believe the equation for this is y=3x-1... would this be helpful in finding the answer and second how did you get the legnths of each side of the triangle. I am having problems trying to find each length. Could someone please go over this problem and answer thank you

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New post 12 Oct 2014, 23:10
garnier77 wrote:
I am having difficulties with this problem I actually got it right but I might have been lucky I tried to find the slope which i believe the equation for this is y=3x-1... would this be helpful in finding the answer and second how did you get the legnths of each side of the triangle. I am having problems trying to find each length. Could someone please go over this problem and answer thank you


Check another discussion of this question: if-a-circle-passes-through-points-1-2-2-5-and-42105.html

Hope it helps.
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Re: M21-20 [#permalink]

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New post 06 Jan 2015, 11:24
Bunuel wrote:
Official Solution:

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\)
B. \(\sqrt{20}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)


Look at the diagram below:

Image

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).


Answer: B



Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

Thanks a ton
Buddy

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New post 07 Jan 2015, 06:17
buddyisraelgmat wrote:
Bunuel wrote:
Official Solution:

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\)
B. \(\sqrt{20}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)


Look at the diagram below:

Image

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).


Answer: B



Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

Thanks a ton
Buddy


Check The Distance Between Two Points here: math-coordinate-geometry-87652.html

Hope it helps.
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Re: M21-20 [#permalink]

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New post 07 Jan 2015, 08:07
Bunuel wrote:
buddyisraelgmat wrote:
Bunuel wrote:
Official Solution:

If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?

A. \(\sqrt{18}\)
B. \(\sqrt{20}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)


Look at the diagram below:

Image

Calculate the lengths of the sides of triangle \(ABC\):

\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\);

\(AC=\sqrt{20}=\sqrt{2}*\sqrt{10}\);

As we see the ratio of the sides of triangle \(ABC\) is \(1:1:\sqrt{2}\), so \(ABC\) is 45°-45°-90° right triangle (in 45°-45°-90° right triangle the sides are always in the ratio \(1:1:\sqrt{2}\)).

So, we have right triangle \(ABC\) inscribed in the circle. Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle , so \(AC=diameter=\sqrt{20}\).


Answer: B



Hi Bunuel

I did not understand how did you calculate AB & BC (highlighted above). Request you to elaborate if possible

Thanks a ton
Buddy


Check The Distance Between Two Points here: math-coordinate-geometry-87652.html

Hope it helps.


Yup. Got it - Thanks

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Re M21-20 [#permalink]

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New post 25 Jul 2015, 12:57
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. How to find the length of each side? Which formula/concept to apply?

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New post 25 Jul 2015, 13:04
rbvigneshwar wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. How to find the length of each side? Which formula/concept to apply?


Check here: if-a-circle-passes-through-points-1-2-2-5-and-42105.html
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Re: M21-20 [#permalink]

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New post 29 Oct 2016, 11:21
Bunuel
I used distance formula using 2 points (1,2) and (5,4) which gave me correct ans. (root {(4-2)^2 + (5-1)^2} )
is this approach fine?

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New post 30 Oct 2016, 00:16
deepak268 wrote:
Bunuel
I used distance formula using 2 points (1,2) and (5,4) which gave me correct ans. (root {(4-2)^2 + (5-1)^2} )
is this approach fine?


Yes, it's a correct way to get the length of AC. The point is how you got that it's a diameter.
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Re: M21-20 [#permalink]

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New post 30 Dec 2016, 02:25
Bunuel,

Tell me if this thought process is ok

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer

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New post 30 Dec 2016, 02:40
Omkar.kamat wrote:
Bunuel,

Tell me if this thought process is ok

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer


x^2 + y^2 = r^2 is the equation of a circle centred at the origin. The given circle is not centred ant the origin. How did you get that \(r = \sqrt{5}\)?

P.S. The correct way is given in the solution above.
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Re: M21-20 [#permalink]

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New post 30 Dec 2016, 02:54
Bunuel wrote:
Omkar.kamat wrote:
Bunuel,

Tell me if this thought process is ok

Equation of a Circle is x^2+y^2= r^2 ...... the points mentioned shud satisfy this equation..we get r= sqrt(5)...diameter= 2*sqrt(5)...if 2 goes inside the sqrt sign, becomes sqrt(20)..thats my answer


x^2 + y^2 = r^2 is the equation of a circle centred at the origin. The given circle is not centred ant the origin. How did you get that \(r = \sqrt{5}\)?

P.S. The correct way is given in the solution above.

Oops...I missed the Origin part of it. Sorry !!

Omkar Kamat
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Re: M21-20 [#permalink]

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New post 14 May 2017, 03:40
Hi Bunuel,

Great solution. However I would like to know how did you realize that finding the lengths of the sides of the triangle formed by the three points would most certainly give you a clue whether this triangle was a right triangle. Frankly, when I started this problem I felt that the only way was to choose an arbitrary point as the centre of circle and equate the distances from the centre to the three points (since they would be radii). This took me a some time. Please share your thoughts.

Regards

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M21-20 [#permalink]

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New post 08 Jul 2017, 03:15
Can also be solved by calculating the slopes of the 2 lines from these 3 points .
One comes to be 3 ie ( 5-2/2-1 = 3) and other as -1/3 ie (4-5/5-2 = -1/3) so there is a right angle and the line connecting the 2 end points will be diameter.

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Isn't solving it by the equation of circle a faster and certain method?

if we hadn't calculated the distances specifically (it doesn't strike naturally to use the distance formula here), we couldn't know then that it is in fact a right angle.

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New post 01 Dec 2017, 06:53
Hi

can anyone explain why triangle ABC is 45-45-90 degree ? i know that one angle must be 90 degrees but the other two angles could be different and all three angles sum is 180. Are sides AB and BC similar ? If so, how ?

Thanks

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New post 01 Dec 2017, 06:56
amargad0391 wrote:
Hi

can anyone explain why triangle ABC is 45-45-90 degree ? i know that one angle must be 90 degrees but the other two angles could be different and all three angles sum is 180. Are sides AB and BC similar ? If so, how ?

Thanks


\(AB=\sqrt{10}\);

\(BC=\sqrt{10}\).

Check The Distance Between Two Points here: http://gmatclub.com/forum/math-coordina ... 87652.html

Hope it helps.
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Re: M21-20   [#permalink] 01 Dec 2017, 06:56
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