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Type: Multiple-Choice
Category: Coordinate Geometry
Level: Grade 10
Standards: HSG-GPE.B.4
Author: nsharp1
Created: 6 years ago

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Coordinate Geometry Question

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Grade 10 Coordinate Geometry CCSS: HSG-GPE.B.4

Ellen is given four points and their coordinates:

$A (-2,0); B (4,4); C (7,3); D(1,-1)$ . She is asked to show whether these points form a parallelogram. In order to do so, she finds the midpoints of

$bar{AC}$ and

$bar{BD}$ . She finds they are both

$(5/2, 3/2)$ . She reasons that, since the midpoints of

$bar{AC}$ and

$bar{BD}$ (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

$ABCD$ is a parallelogram. Is she correct, and why?

No, she made a calculation error, and the midpoints of $bar{AC}$ and $bar{BD}$ 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 $\stackrel{leftrightarrow}{AC}$ and $\stackrel{leftrightarrow}{BD}$ , find their intersection point, $P$ , and then see if $bar{AP}, bar{PC}$ are congruent, and then if $bar{BP}, bar{DP}$ are congruent.
Yes, this sufficiently shows that $ABCD$ is a parallelogram.