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# M21-20

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Math Expert
Joined: 02 Sep 2009
Posts: 49858

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16 Sep 2014, 01:11
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Difficulty:

25% (medium)

Question Stats:

78% (01:33) correct 22% (01:50) wrong based on 133 sessions

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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. $$\sqrt{30}$$

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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16 Sep 2014, 01:11
1
3
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:

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 $$AC=diameter=\sqrt{20}$$.

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Intern
Joined: 26 Jul 2014
Posts: 1

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12 Oct 2014, 20:55
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
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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13 Oct 2014, 00:10
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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Manager
Joined: 14 Jul 2014
Posts: 93

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06 Jan 2015, 12:24
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:

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 $$AC=diameter=\sqrt{20}$$.

Hi Bunuel

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

Thanks a ton
Buddy
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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07 Jan 2015, 07:17
1
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:

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 $$AC=diameter=\sqrt{20}$$.

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.
_________________
Manager
Joined: 14 Jul 2014
Posts: 93

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07 Jan 2015, 09:07
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:

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 $$AC=diameter=\sqrt{20}$$.

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
Intern
Joined: 09 Apr 2013
Posts: 1
Concentration: Nonprofit, Entrepreneurship
GMAT Date: 09-15-2015

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25 Jul 2015, 13:57
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?
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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25 Jul 2015, 14:04
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
_________________
Intern
Joined: 27 Feb 2015
Posts: 48
Concentration: General Management, Economics
GMAT 1: 630 Q42 V34
WE: Engineering (Transportation)

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29 Oct 2016, 12:21
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?
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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30 Oct 2016, 01:16
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.
_________________
Intern
Joined: 04 Mar 2016
Posts: 45
Location: India

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30 Dec 2016, 03:25
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
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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30 Dec 2016, 03:40
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.
_________________
Intern
Joined: 04 Mar 2016
Posts: 45
Location: India

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30 Dec 2016, 03:54
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
When The Going Gets Tough, The Tough Gets Going !!
Intern
Joined: 05 Dec 2016
Posts: 8

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14 May 2017, 04:40
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
Manager
Joined: 20 Jun 2014
Posts: 52
GMAT 1: 630 Q49 V27
GMAT 2: 660 Q49 V32

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08 Jul 2017, 04:15
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.
Intern
Joined: 10 Feb 2017
Posts: 1

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12 Jul 2017, 13:21
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.
Intern
Joined: 09 Sep 2015
Posts: 24

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01 Dec 2017, 07:53
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
Math Expert
Joined: 02 Sep 2009
Posts: 49858

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01 Dec 2017, 07:56
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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Intern
Joined: 24 Jun 2016
Posts: 3

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09 Apr 2018, 15:23
Why go through the trouble of pythag. theorem and finding separate leg lengths when you can just use distance formula between A-C (easy to surmise that distance AC is greater than AB or BC).
Is there a possibility that the GMAT just throws you three arbitrary points on the circle as opposed to this equation that included the endpoints of the diameter?
Re: M21-20 &nbs [#permalink] 09 Apr 2018, 15:23
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# M21-20

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