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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 (-1,3) with radius 3, which of the following is true about the point (0.5,0.2)?
  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: 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 ¯AC and ¯BD. She finds they are both (52,32). She reasons that, since the midpoints of ¯AC and ¯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?
  1. No, she made a calculation error, and the midpoints of ¯AC and ¯BD 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 ↔AC and ↔BD, find their intersection point, P, and then see if ¯AP,¯PC are congruent, and then if ¯BP,¯DP are congruent.
  4. Yes, this sufficiently shows that ABCD is a parallelogram.

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