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Difficulty: Very hardGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a rhombus ABCDABCD has vertices A(−2,3)A(-2, 3) and C(4,1)C(4, 1). If vertex BB lies on the line y=2x−5y = 2x - 5, what is the area of the rhombus ABCDABCD?

  1. A
    20
  2. B
    40
  3. C
    80
  4. 100Answer
  5. E
    200

Answer

100
The correct answer is 100. By finding the midpoint of diagonal ACAC as (1,2)(1, 2) and using the perpendicular slope of 33, the line containing diagonal BDBD is y=3x−1y = 3x - 1. Intersecting this with y=2x−5y = 2x - 5 yields vertex B(−4,−13)B(-4, -13), which gives vertex D(6,17)D(6, 17) by midpoint symmetry. The diagonal lengths are 2102\sqrt{10} and 101010\sqrt{10}, and their half-product is 100.

Step-by-Step Solution

1
Find the midpoint MM of diagonal ACAC.
M(1,2)M(1, 2)
Since ABCDABCD is a rhombus, its diagonals bisect each other at their midpoint.
2
Calculate the slope of diagonal ACAC.
mAC=−13m_{AC} = -\frac{1}{3}
The slope is calculated using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} with vertices A(−2,3)A(-2, 3) and C(4,1)C(4, 1).
3
Find the equation of the line containing diagonal BDBD.
y=3x−1y = 3x - 1
The diagonals of a rhombus are perpendicular. The slope of BDBD is the negative reciprocal of −13-\frac{1}{3}, which is 33. The line passes through M(1,2)M(1, 2).
4
Find the coordinates of vertex BB.
B(−4,−13)B(-4, -13)
Vertex BB lies at the intersection of the diagonal line y=3x−1y = 3x - 1 and the given line y=2x−5y = 2x - 5.
5
Find the coordinates of vertex DD.
D(6,17)D(6, 17)
Since M(1,2)M(1, 2) is the midpoint of diagonal BDBD, we solve −4+xD2=1\frac{-4 + x_D}{2} = 1 and −13+yD2=2\frac{-13 + y_D}{2} = 2.
6
Calculate the lengths of the diagonals ACAC and BDBD.
AC=210AC = 2\sqrt{10} and BD=1010BD = 10\sqrt{10}
Use the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} on the pairs of vertices.
7
Calculate the area of the rhombus.
100100
The area of a rhombus is given by Area=12×d1×d2\text{Area} = \frac{1}{2} \times d_1 \times d_2.

Key Concept

Using perpendicular bisector properties of rhombus diagonals on a coordinate plane to find vertices and calculate area.
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Topics

Geometric Figures on the Coordinate Plane

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