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Re: Area of region [#permalink]
15 Jan 2012, 09:26

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Apex231 wrote:

Please move this to PS thread...

If equation |x/2|+|y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A 20 B 50 C 100 D 200 E 400

First of all to simplify the given expression a little bit let's multiply it be 2: \(|\frac{x}{2}|+|\frac{y}{2}|=5\) --> \(|x|+|y|=10\).

Now, find x and y intercepts of the region (x-intercept is a value(s) of x for y=0 and similarly y-intercept is a value(s) of y for x=0): \(y=0\) --> \(|x|=10\) --> \(x=10\) and \(x=-10\); \(x=0\) --> \(|y|=10\) --> \(y=10\) and \(y=-10\).

So we have 4 points: (10, 0), (-10, 0), (0, 10) and (-10, 0).

When you join them you'll get the region enclosed by \(|x|+|y|=10\):

Attachment:

Enclosed region.gif [ 2.04 KiB | Viewed 7234 times ]

You can see that it's a square. Why a square? Because diagonals of the rectangle are equal (20 and 20), and also are perpendicular bisectors of each other (as they are on X and Y axis), so it must be a square. As this square has a diagonal equal to 20, so the \(Area_{square}=\frac{d^2}{2}=\frac{20*20}{2}=200\).

Or the \(Side= \sqrt{200}\) --> \(area=side^2=200\).

If equation |x/2|+|y/2| = 5 encloses a certain region [#permalink]
10 Sep 2012, 11:25

CMcAboy wrote:

Can someone help me with this question:

If equation |x/2| + |y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A) 20 B) 50 C) 100 D) 200 E) 400

I believe this is the simplest & the quickest solution |x/2| + |y/2| = 5 Put x = 0 in the above equation we get |y/2| = 5, which means y= 10, - 10 Put y = 0 in the above equation we get |y/2| = 5, which means x= 10, - 10

If you see plot these four points you get a square with two equal diagonals of length 20 units Thus area = 1/2 * (Diagonal)^2 -----> 1/2 * 400 = 200

I hope this will help many. _________________

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Re: If equation |x/2|+|y/2| = 5 encloses a certain region [#permalink]
12 Oct 2014, 02:26

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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If equation |x/2|+|y/2| = 5 encloses a certain region [#permalink]
15 Oct 2014, 18:17

Apex231 wrote:

If equation |x/2|+|y/2| = 5 encloses a certain region on the coordinate plane, what is the area of this region?

A. 20 B. 50 C. 100 D. 200 E. 400

Hello There, Equation of a straight line whose x and y intercepts are a and b resp. is (x/a) + (y/b) = 1 i.e., coordinates of two ends of the line are (a,0) and (0,b). Now, from the given question, |x/2|+|y/2| = 5, reducing this to intercept form we get, |x/10|+|y/10| = 1 Considering the equation without modulus, coordinates are (10,0) and (0,10). Since there is modulus, other two coordinates are (-10,0) and (0,-10). Now coordinates (10,0), (0,10), (-10,0) and (0,-10) form a square with diagonal length = 20. Here diagonal length can be obtained by calculating the distance between (10,0) and (-10,0) or (0,10) and (0,-10). In a square, Diagonal = Side * sqrt(2) Side = 10 * sqrt(2) Area = Side * Side = 200.

Ans : D

Hope this helps! Thanks! _________________

Regards, Bharat Bhushan Sunkara.

"You need to sacrifice what you are TODAY, for what you want to be TOMORROW!!"

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