(a) Sketch the parallelogram with vertices A(-2,-1) B(4,2),...

(a) Sketch the parallelogram with vertices $A(-2,-1)$ $B(4,2), C(7,7),$ and $D(1,4) .$
(b) Find the midpoints of the diagonals of this parallelogram.
(c) From part (b), show that the diagonals bisect each other.

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Key Concepts

Parallelogram

A parallelogram is a four-sided polygon (quadrilateral) with both pairs of opposite sides parallel. Its properties include congruent opposite sides, congruent opposite angles, and the fact that its diagonals bisect each other. This shape is fundamental in geometry and serves as a basis for understanding vector addition and affine transformations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves representing geometric figures using coordinates on a plane. This approach allows for the precise calculation of distances, midpoints, and slopes, and it is essential for graphically constructing shapes like parallelograms from given vertices.

Midpoint Formula

The midpoint formula is used to find the exact center point of a line segment in the coordinate plane by averaging the x-coordinates and the y-coordinates of its endpoints. It is crucial in determining the midpoints of the diagonals, which in turn demonstrates properties like the bisection property in parallelograms.

Diagonal Bisectors

One of the key properties of parallelograms is that their diagonals bisect each other; this means that the point where the diagonals intersect divides each diagonal into two segments of equal length. This property is often used as a method to verify that a given quadrilateral is a parallelogram.

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