BTGmoderatorDC 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:
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.
The correct answer is
B.
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