Question

Prove using coordinate geometry: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Answer 1

Could I ask why you reposted this?

Answer 2

yep i got that what they have given what i have to prove its confusing

Answer 3

If a point is equidistant from the endpoints of the segment then it is on the perpendicular bisector of the segment.



Let [AB] be a given segment. Let M be an arbitrary point in space such that AM = BM.

Project M onto (d) and call H the intersection point of (d) and (AB).

Consider the triangles AMH and BMH.

These two triangles have the angle HBM = the angle HAM (Because the triangle ABM is isosceles in M), and the angle BHM = the angle AHM (= 90 degrees). Therefore, the angle AMH = the angle BMH (two of the angles are the same in both triangles, so the other must be the same)

These triangles also have AM = BM, they also have a common side which is [MH].

In summary, we showed that the triangles have two equal sides, and the angles between those sides are also equal. Therefore these triangles are congruent. Therefore AH = BH and thus H is the midpoint of [AB]. Hence, (d) is on the perpendicular bisector of the [AB].

Answer 4

i gave my teacher this answer and she says its incomplete

Answer 5

It is not a coordinate geometry proof.

Answer 6

Coordinate geometry proofs use coordinates to prove them.

Answer 7

the proof will be completed in two steps.
1) prrove that if P is on the bisector it is equidistant
2) now, proove that if it is equidistant, it HAS to be on the perpendicular bisector!!

Answer 8

Referring to the diagram I drew previously... the distance from P to A is :

Answer 9

\[\sqrt{(0--a)^2+(c-0)^2}=\sqrt{a^2+c^2}\]

Answer 10

Similarly, the distance from P to B is:
\[PB=\sqrt{(0-a)^2+(c-0)^2}=\sqrt{a^2+c^2}\]

Answer 11

brb

Answer 12

@electrokid Thanks!!

Answer 13

@Mertsj seems that this question was asked not twice but thrice by the same person on the same day within 20min!!!!

Answer 14

but the problem is that we have to give reasons regarding each one

Answer 15

http://openstudy.com/study#/updates/5158508ae4b07077e0c08709