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Author: nsharp1
No. Questions: 3
Created: Nov 6, 2018
Last Modified: 2 years ago

Points on perpendicular bisector proof

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Given line segment [math]bar{AC}[/math] with midpoint [math]B[/math] and perpendicular bisector [math]l[/math] (not pictured), which intersects [math]bar{AC}[/math] at [math]B[/math], prove that any point on line [math]l[/math] is equidistant from points [math]A[/math] and [math]C[/math].

Segment ABC

[math] " Statement " [/math][math] " Reason "[/math]
[math]1. "Let " P " be an arbitrary point on line " [/math] [math] \ \ \ \ \ l " that does not lie on "bar{AC}[/math][math]1. "A line consists of an infinite number of points"[/math]
[math]2. "Construct "bar{AP}" and "bar{CP}[/math][math]2. "Two points define a line segment"[/math]
[math]3. [/math][math]3. "Reflexive property"[/math]
[math]4. B " is the midpoint of " bar{AC}[/math][math]4. "Given" [/math]
[math]5. bar{AB} ~= bar{BC} [/math][math]5. "Definition of midpoint"[/math]
[math]6. l " is the perpendicular bisector of " bar{AC} [/math][math]6. "Given"[/math]
[math]7. ang ABP " and " ang PBC " are right angles" [/math][math]7. "Definition of perpendicular"[/math]
[math]8. ang ABP ~= ang PBC [/math][math]8. "All right angles are congruent"[/math]
[math]9. Delta ABP ~= Delta CBP [/math][math]9. ""[/math]
[math]10. bar{AP} ~= bar{PC} [/math][math]10. "Corresponding sides of congruent triangles " [/math] [math] \ \ \ \ " are congruent"[/math]
[math]11. P " is equidistant from " A " and " C [/math][math]11. "Definition of equidistant"[/math]
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
What is the missing statement in step 3?
  1. [math]l ~= l [/math]
  2. [math]bar{BP} ~= bar{BP[/math]
  3. [math]bar{AB} ~= bar{BC} [/math]
  4. [math]bar{AC} ~= bar{AC}[/math]
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9
What still needs to be done in order to complete the proof?
  1. Consider other points on [math]l[/math], showing the same thing holds true for these other points.
  2. Redraw the diagram, showing that the results are still true if [math]l[/math] is horizontal and [math]bar{AC}[/math] is vertical.
  3. State that, if the point on [math]l[/math] lies on [math]bar{AC}[/math], that it is equidistant from points [math]A[/math] and [math]B[/math], due to [math]l[/math] bisecting [math]bar{AC}[/math] (from given information).
  4. Nothing, it is complete.