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# If points A and B are 5 units apart, which of the following

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Senior Manager
Joined: 31 Oct 2011
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If points A and B are 5 units apart, which of the following [#permalink]

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31 Mar 2012, 02:29
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If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 39702
Re: If points A and B are 5 units apart, which of the following [#permalink]

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04 Dec 2014, 06:52
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arnabs wrote:
bunuel,
III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

because, the diagonal AB = 5, you have assumed the other two sides to be 3 and 4 for the rectangle. this is what i am wanting to clarify.

The question asks which of the following can have only one possible value?

III suggests that it's the area of a rectangular region with diagonal AB:

AB could be the diagonal of a square as well as it could be the diagonal of a rectangle with the sides equal to 3 and 4 (of course there are many other cases possible). Those two cases give different areas for a rectangle with diagonal of 5. Therefore, the area of a rectangular region with diagonal of 5 can have more than one value.

Hope it's clear.
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Re: If points A and B are 5 units apart, which of the following [#permalink]

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31 Mar 2012, 02:42
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eybrj2 wrote:
If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

I. The area of a square region with side AB --> side=AB=5 --> the area of a square is 5^2=25, so only one possible value;

II. The circumference of a circle passing through points A and B --> circle is not defined by only two points, hence the circumference of a circle passing through points A and B can take infinitely many values. For example consider when AB=diameter and when points A and B are not the endpoints of the diameter;

III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

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Re: If points A and B are 5 units apart, which of the following [#permalink]

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31 Mar 2012, 02:46
eybrj2 wrote:
If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

1: Area od square = (Side)^2=(AB)^2=25 ( fixed )
2: radius of circle can be anything as center of circle is pivot.==> radius is not fix ==> 2*pi*r is not fix.
3. As AB as diagonal is fix ==> point A and B are fix ==> Other point C and D of rectangle are not fix ==> Sides are not fix ==> Area is not fix.

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Re: If points A and B are 5 units apart, which of the following [#permalink]

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28 Jul 2014, 06:59
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Re: If points A and B are 5 units apart, which of the following [#permalink]

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04 Dec 2014, 06:37
Bunuel wrote:
eybrj2 wrote:
If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

I. The area of a square region with side AB --> side=AB=5 --> the area of a square is 5^2=25, so only one possible value;

II. The circumference of a circle passing through points A and B --> circle is not defined by only two points, hence the circumference of a circle passing through points A and B can take infinitely many values. For example consider when AB=diameter and when points A and B are not the endpoints of the diameter;

III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

Bunuel,
i have a doubt here. from one of your previous posts, i remember you saying that in a right angled triangle, if its not explicitly mentioned that the legs of the triangle have values that are integers, then we cant apply the 3^2 + 4^2 = 5^2 rule.

in the rectangle when its mentioned that AB is a diagonal, then the legs of the right angled triangle so fromed inside the rectangle may or may not have the values of its legs as 3 and 4.

how i approached this problem,
statement 1 is correct
statement 2 mentions that a circle passed through AB. now AB may or may not be the radius of the circle. hence insuff
statement 3 (keeping your sugesstion in mind) : diagonal AB=5, hence if sides are 3 and 4, area is 12. if not 3 and 4, area will be soemthing else. hence, insuff.

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Posts: 39702
Re: If points A and B are 5 units apart, which of the following [#permalink]

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04 Dec 2014, 06:40
arnabs wrote:
Bunuel wrote:
eybrj2 wrote:
If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

I. The area of a square region with side AB --> side=AB=5 --> the area of a square is 5^2=25, so only one possible value;

II. The circumference of a circle passing through points A and B --> circle is not defined by only two points, hence the circumference of a circle passing through points A and B can take infinitely many values. For example consider when AB=diameter and when points A and B are not the endpoints of the diameter;

III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

Bunuel,
i have a doubt here. from one of your previous posts, i remember you saying that in a right angled triangle, if its not explicitly mentioned that the legs of the triangle have values that are integers, then we cant apply the 3^2 + 4^2 = 5^2 rule.

in the rectangle when its mentioned that AB is a diagonal, then the legs of the right angled triangle so fromed inside the rectangle may or may not have the values of its legs as 3 and 4.

how i approached this problem,
statement 1 is correct
statement 2 mentions that a circle passed through AB. now AB may or may not be the radius of the circle. hence insuff
statement 3 (keeping your sugesstion in mind) : diagonal AB=5, hence if sides are 3 and 4, area is 12. if not 3 and 4, area will be soemthing else. hence, insuff.

I don't understand where you see contradiction in what I've written. Please re-read the question and the solution.
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Re: If points A and B are 5 units apart, which of the following [#permalink]

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04 Dec 2014, 06:47
bunuel,
III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

because, the diagonal AB = 5, you have assumed the other two sides to be 3 and 4 for the rectangle. this is what i am wanting to clarify.
Manager
Joined: 06 Aug 2013
Posts: 92
Re: If points A and B are 5 units apart, which of the following [#permalink]

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04 Dec 2014, 06:57
Bunuel wrote:
arnabs wrote:
bunuel,
III. The area of a rectangular region with diagonal AB --> consider the case when AB is the diagonal of a square (the area is 25/2) and the case when AB is the diagonal of a rectangle with the sides equal to 3 and 4 (the area 12).

because, the diagonal AB = 5, you have assumed the other two sides to be 3 and 4 for the rectangle. this is what i am wanting to clarify.

The question asks which of the following can have only one possible value?

III suggests that it's the area of a rectangular region with diagonal AB:

AB could be the diagonal of a square as well as it could be the diagonal of a rectangle with the sides equal to 3 and 4 (of course there are many other cases possible). Those two cases give different areas for a rectangle with diagonal of 5. Therefore, the area of a rectangular region with diagonal of 5 can have more than one value.

Hope it's clear.

yeah, exactly... that is what i wanted to confirm. your approach was one of the approaches of solving, just making sure that i am approaching problems like these in the right fashion. 6 days for the D-day, hence, a little freaked out. thanks for the confirmation bunuel, kudos to you.
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Re: If points A and B are 5 units apart, which of the following [#permalink]

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02 Nov 2016, 06:51
eybrj2 wrote:
If points A and B are 5 units apart, which of the following can have only one possible value?

I. The area of a square region with side AB
II. The circumference of a circle passing through points A and B
III. The area of a rectangular region with diagonal AB

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

1. always true. so B, C are out immediately.
2. we can have multiple options...AB might be a chord, or AB might be the diameter. D is out.
we are left with A and E...pretty good odds, hah?
3. diagonal is AB -> since it's a rectangular region, we might have the rectangular formed by 2 triangles 3-4-5. since we are not 100% sure of that, we can't have a definite answer. E is out.

Re: If points A and B are 5 units apart, which of the following   [#permalink] 02 Nov 2016, 06:51
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