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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 $bar{AC}$ 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?
1. No, she made a calculation error, and the midpoints of $bar{AC}$ and $bar{BD}$ 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 $\stackrel{leftrightarrow}{AC}$ 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.
4. Yes, this sufficiently shows that $ABCD$ is a parallelogram.