Carcass 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.
Quantity A |
Quantity B |
the area of rectangle ABCD |
30 |
A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
If AC is the diagonal, and lies on the x-axis, we know that point C = (-5,0). In order to calculate the B point, we have to find the next intersection between the line and circle:
\(x^2 + y^2 = 25\)
\(x^2 + (3x+15)^2 = 25\)
\(10x^2 + 50x + 200 = 0\)
\(x^2 + 5x + 20 = 0\)
Then, we know that the roots are:
\(x=-4 or x =-5\)
We already know that -5 is one of the points, therefore, we have to evaluate -4.
Evaluating x=-4 in y=3x+15 yields y=3.
Having said that, the side BC is \(1^2 + 3^2 = BC^2\), \(BC = \sqrt{10}\)
Now, we can use the pythagorean theorem in order to calculate side AB. \(AB^2 + \sqrt{10}^2 = 10^2\), \(AB = \sqrt{90}\).
Finally, the area is
\(A = \sqrt{90}\sqrt{10}\)
\(A = \sqrt{900}\)
\(A = 30\)
Option C.
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