In the xy-coordinate system, rectangle ABCD is inscribed

[BTGmoderatorLU]

Source: GMAT Prep

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

The OA is B

[Jay@ManhattanReview]

Source: GMAT Prep

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

The OA is B

Since the equation of the circle is x^2 + y^2 = 25, its radius is √25 = 5 and the diameter is 2*5 = 10.

Since vertex B lies on the x^2 + y^2 = 25 as well as on y = 3x + 15, the x and y coordinates of vertex B would satisfy both the equations.

Plugging-in y = 3x + 15 in x^2 + y^2 = 25, we get coordinates of vertex B: (-5, 0) and (-4, 3). Since B lies in II quadrant, coordinate of vertex B are (-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; taking AC as base and perpendicular from vertex B on X-axis as height

=> Area of the rectangle = 2 * Area of the triangle ABC = 2 * 15 = 30

The correct answer: B

Hope this helps!

-Jay
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