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In the xy-coordinate system, rectangle ABCD is inscribed

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In the xy-coordinate system, rectangle ABCD is inscribed [#permalink]

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24 Mar 2005, 00:37
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In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x^2+y^2=25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y=3x+15 , what is the area of rectangle ABCD?

(A) 15
(B) 30
(C) 40
(D) 45
(E) 50

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24 Mar 2005, 02:44
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30

Since the equation of the circle x^2+y^2=25, it represents a circle with centre at the origin and radius 5.

The explaination about the rectagle suggests that one of the diagonals is the diameter of the circle. Therefore, the two diametrically opposite points on the X axis would be (-5,0) and (5,0) the length of the diameter being 10.

Now, to find out about the location of point B. Since it lies on the circle, it should satisfy the equation x^2+y^2=25. Moreover, since it lies on the line y=3x+15, it should satisfy this equation as well. Solving these two equations, we get the possible values of B to be (-5, 0) and (-4, 3). Since B lies in the II quadrant, we take B to be (-4, 3).

Area of the rectangle ABCD = 2 * area of the triangle ABC
Area of triangle ABC = 1/2 * base * height = 1/2 * 10 * 3 = 15
=> Area of the rectangle = 2 * 15 = 30

Edit 1: calcuation mistake.
Edit 2: Attached a file displaying the problem pictorially.
Attachments

File comment: Figure for explaination
PS_Geometry.doc [25.5 KiB]

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24 Mar 2005, 10:29
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The only thing required is the 'y' co-ordinate of (B), which can be easily found by substituting the value of 'x' from y = 3x + 15 ==> x = (y-15)/3

into equation of circle. It gives y = 3 and y = 0. Obviously y = 0 is the corordiante for 'C'.

Now diagonal is the base of the 'half-rectangle', whose height is '3' (we just found). This diagonal is the diameter of circle = 10

Hence area of triangle = 1/2 * 10 * 3

Hence area of rectangle = 2 * area of triangle = 30

Ketan

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25 Mar 2005, 05:33
any other takers ???

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27 Mar 2005, 02:29
both of you are right, the answer is 30.

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Re: In the xy-coordinate system, rectangle ABCD is inscribed [#permalink]

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02 Sep 2015, 08:55
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Re: In the xy-coordinate system, rectangle ABCD is inscribed [#permalink]

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20 Feb 2017, 15:22
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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Re: In the xy-coordinate system, rectangle ABCD is inscribed   [#permalink] 20 Feb 2017, 15:22
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