Coordinate Geometry Question
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Grade 10 Coordinate Geometry CCSS: HSG-GPE.B.4
Ellen is given four points and their coordinates: [math]A (-2,0); B (4,4); C (7,3); D(1,-1)[/math]. She is asked to show whether these points form a parallelogram. In order to do so, she finds the midpoints of [math]bar{AC}[/math] and [math]bar{BD}[/math]. She finds they are both [math](5/2, 3/2)[/math]. She reasons that, since the midpoints of [math]bar{AC}[/math] and [math]bar{BD}[/math] (the diagonals of the quadrilateral) are coincident, these line segments intersect at each other's midpoints and thus bisect each other. Therefore, she concludes that [math]ABCD[/math] is a parallelogram. Is she correct, and why?
- No, she made a calculation error, and the midpoints of [math]bar{AC}[/math] and [math]bar{BD}[/math] are not the same.
- No, she must show that both sets of opposite sides are parallel.
- No, her reasoning is incorrect. She must find the equations of lines [math]\stackrel{leftrightarrow}{AC}[/math] and [math]\stackrel{leftrightarrow}{BD}[/math], find their intersection point, [math]P[/math], and then see if [math]bar{AP}, bar{PC}[/math] are congruent, and then if [math]bar{BP}, bar{DP}[/math] are congruent.
- Yes, this sufficiently shows that [math]ABCD[/math] is a parallelogram.
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