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Common Core Standard HSG-GPE.B.4 Questions

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

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Grade 10 Coordinate Geometry CCSS: HSG-GPE.B.4
For the circle centered at [math](-1,3)[/math] with radius 3, which of the following is true about the point [math](0.5, 0.2) ?[/math]
  1. It lies on the circle.
  2. It lies in the circle.
  3. It lies outside the circle.
  4. Not enough information.
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?
  1. No, she made a calculation error, and the midpoints of [math]bar{AC}[/math] and [math]bar{BD}[/math] are not the same.
  2. No, she must show that both sets of opposite sides are parallel.
  3. 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.
  4. Yes, this sufficiently shows that [math]ABCD[/math] is a parallelogram.

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