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In the xy-coordinate system, rectangle ABCD is inscribed within a circ

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

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New post 04 Apr 2018, 03:54
2
gmatmo wrote:

thanks for this awesome approach, but how do you know that -4,3 has to lie on the circle, why isn't it (-3,6)?


The distance of any point lying on the circle from (0, 0) must be 5 (since the radius of the circle is 5).
(-4, 3) is at a distance of 5 from (0, 0) but (-3, 6) is not (even though it lies on the line). It will not lie on the circle.
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Re: In the xy-coordinate system, rectangle ABCD is inscribed within a circ  [#permalink]

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New post 15 May 2019, 02:47
AnkitK wrote:
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


x²+y² = r² is the equation for a circle with its center at the origin and a radius of r.
Thus, x²+y² = 25 is a circle with a center at the origin and a radius of 5.
The information in the prompt yields the following figure:
Image

OB is a radius and thus has a length of 5.
Given the GMAT's love of special triangles, it is likely that triangle BOE is a 3-4-5 triangle, implying that B is located either at (-3, 4) or (-4, 3).
B lies on the line y=3x+15.
If we plug x=-4 into y=3x+15, we get y=3, implying that (-4, 3) lies on y=3x+15.
Thus, B is located at (-4, 3).

Since BE=3, the area of triangle ABC = (1/2)bh = (1/2)(CA)(BE) = (1/2)(10)(3) = 15.
Since the triangle ABC is 1/2 of rectangle ABCD, we get:
ABCD=30.


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

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New post 10 Aug 2019, 09:00
We don't actually need to find the value of AB BC and AC. Since Point B is perpendicular to AC the area of the rectangle can simply be 3*10=30
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In the xy-coordinate system, rectangle ABCD is inscribed within a circ  [#permalink]

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New post 24 Sep 2019, 09:11
AnkitK wrote:
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


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?

Point B
x^2 + (3x+15)^2 = 25
x^2 + 9x^2 + 90x +225 = 25
10x^2 + 90x + 200 = 0
x^2 + 9x + 20 = 0
(x+5)(x+4) = 0
x = -5 or x = -4
(x,y) = {(-5,0),(-4,3)}
Since point B lies in II quadrant
(x,y) = (-4,3)
Area of ABC = 1/2 * 10 * 3 = 15
Area of ABCD = 15*2 = 30

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

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New post 21 Dec 2019, 07:44
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Re: In the xy-coordinate system, rectangle ABCD is inscribed   [#permalink] 21 Dec 2019, 07:44

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