Points, Lines, and Planes Question

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Given line segment

$bar{AC}$ with midpoint

$B$ and perpendicular bisector

$l$ (not pictured), which intersects

$bar{AC}$ at

$B$ , prove that any point on line

$l$ is equidistant from points

$A$ and

$C$ .

$" Statement "$	$" Reason "$
$1. "Let " P " be an arbitrary point on line "$ $\ \ \ \ \ l " that does not lie on "bar{AC}$	$1. "A line consists of an infinite number of points"$
$2. "Construct "bar{AP}" and "bar{CP}$	$2. "Two points define a line segment"$
$3.$	$3. "Reflexive property"$
$4. B " is the midpoint of " bar{AC}$	$4. "Given"$
$5. bar{AB} ~= bar{BC}$	$5. "Definition of midpoint"$
$6. l " is the perpendicular bisector of " bar{AC}$	$6. "Given"$
$7. ang ABP " and " ang PBC " are right angles"$	$7. "Definition of perpendicular"$
$8. ang ABP ~= ang PBC$	$8. "All right angles are congruent"$
$9. Delta ABP ~= Delta CBP$	$9. ""$
$10. bar{AP} ~= bar{PC}$	$10. "Corresponding sides of congruent triangles "$ $\ \ \ \ " are congruent"$
$11. P " is equidistant from " A " and " C$	$11. "Definition of equidistant"$

Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9

What still needs to be done in order to complete the proof?

Consider other points on $l$ , showing the same thing holds true for these other points.
Redraw the diagram, showing that the results are still true if $l$ is horizontal and $bar{AC}$ is vertical.
State that, if the point on $l$ lies on $bar{AC}$ , that it is equidistant from points $A$ and $B$ , due to $l$ bisecting $bar{AC}$ (from given information).
Nothing, it is complete.

Question Info

Points, Lines, and Planes Question

Grade 10 Points, Lines, and Planes CCSS: HSG-CO.C.9