In the xy-coordinate system, rectangle ABCD is inscribed

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

OA B

Source: GMAT Prep

The equation of the circle x^2 + y^2 = 25 suggests that the given circle has its center at the origin and radius equal to √25 = 5. Thus, AD = 2*5 = 10 = Diameter. Since AD lies on the X-axis, the coordinate of A = (–5, 0) and the coordinate of D = (5, 0).

Now we are given that vertex B lies on the quadrant II and n the circle. Thus, the point must satisfy the equation x^2 + y^2 = 25 and the line y = 3x + 15.

Plugging in the value of y = 3x + 15 in x^2 + y^2 = 25, we get x = –5 or –4; the corresponding value of y are 0 and 3, respectively. Thus, B can be (–5, 0) or (–4, 3). Again, the vertex B lies on the quadrant II, the coordinate of B would be (–4, 3).

Area of the rectangle ABCD = 2 * Area of the triangle ABC
Area of the triangle ABC = 1/2 * (Perpendicular dropped from vertex B on AD) * (Base AD) = 1/2 * (Y-coordinate of vertex B) * 10 = 1/2 * 3 * 10 = 15

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

The correct answer: B

Hope this helps!

-Jay
_________________
Manhattan Review GMAT Prep

Locations: GRE Manhattan | ACT Tutoring Houston | SAT Prep Courses Seattle | Charlotte IELTS Tutoring | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.