(a) Sketch the parallelogram with vertices A(-7,-1) B(4,3),...

(a) Sketch the parallelogram with vertices $A(-7,-1)$ $B(4,3), Q(7,8),$ and $D(-4,4).$
(b) Compute the midpoints of the diagonals $\overline{A C}$ and $\overline{B D}$.
(c) What conclusion can you draw from part (b)?

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Key Concepts

Parallelogram Properties

A parallelogram is a quadrilateral with two pairs of parallel sides. One key property of parallelograms is that the diagonals bisect each other, meaning each diagonal cuts the other into two equal parts. This property is fundamental when analyzing figures in coordinate geometry as it provides a method to verify if a quadrilateral is a parallelogram.

Midpoint Formula

The midpoint formula is used in coordinate geometry to find the point exactly halfway between two given points. It is computed by averaging the corresponding coordinates of the endpoints. This formula is crucial in many problems involving segments and shapes, such as determining the common point where the diagonals of a parallelogram intersect.

Diagonals Bisecting Themselves

The result that the midpoints of the diagonals of a parallelogram are the same demonstrates the property that a parallelogram's diagonals bisect each other. This fact not only confirms the shape's classification as a parallelogram but also serves as a key verification tool in coordinate geometry problems involving quadrilaterals.

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