Given the coordinates of points $A(-2, 1)$, $B(1, 0)$, and $C(2, 4)$, find the coordinates of point $E$ such that quadrilateral $ABEC$ is a parallelogram.

GeometryCoordinate GeometryVectorsParallelogramsGeometric Proof
2025/5/11

1. Problem Description

Given the coordinates of points A(2,1)A(-2, 1), B(1,0)B(1, 0), and C(2,4)C(2, 4), find the coordinates of point EE such that quadrilateral ABECABEC is a parallelogram.

2. Solution Steps

In a parallelogram, opposite sides are parallel and equal in length. In parallelogram ABECABEC, we have ABCEAB \parallel CE and ACBEAC \parallel BE. Also, AB=CE\vec{AB} = \vec{CE} and AC=BE\vec{AC} = \vec{BE}.
Let E=(x,y)E = (x, y). We can use AB=CE\vec{AB} = \vec{CE} to find the coordinates of EE.
AB=BA=(1(2),01)=(3,1)\vec{AB} = B - A = (1 - (-2), 0 - 1) = (3, -1)
CE=EC=(x2,y4)\vec{CE} = E - C = (x - 2, y - 4)
Since AB=CE\vec{AB} = \vec{CE}, we have:
x2=3x - 2 = 3 and y4=1y - 4 = -1
x=3+2=5x = 3 + 2 = 5
y=1+4=3y = -1 + 4 = 3
Therefore, E=(5,3)E = (5, 3).
We can also use AC=BE\vec{AC} = \vec{BE}.
AC=CA=(2(2),41)=(4,3)\vec{AC} = C - A = (2 - (-2), 4 - 1) = (4, 3)
BE=EB=(x1,y0)\vec{BE} = E - B = (x - 1, y - 0)
Since AC=BE\vec{AC} = \vec{BE}, we have:
x1=4x - 1 = 4 and y0=3y - 0 = 3
x=4+1=5x = 4 + 1 = 5
y=3y = 3
Therefore, E=(5,3)E = (5, 3).

3. Final Answer

The coordinates of point EE are (5,3)(5, 3).

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