okay i didn't really have much room in the question to make it perfectly clear: Prove that if a point lies on the two ends of a line segment, the point lies on the perpendicular bisector of that segment. I get why, I just don't know how to prove it. Thanks so much. Even just a start would be great!
2007-08-31 09:14:30 · 7 answers · asked by whaddyaknow? 4 in Science & Mathematics ➔ Mathematics
sorry i meant to say if the the point is equidistant from the two ends, not lies on the two ends. stupid typos.
2007-08-31 09:27:18 · update #1
You want a PROOF, not just statements that "it must be!" Here is that proof:
Let the line segment be AB, and the point equidistant from its ends be P.
Drop a perpendicular from P to AB, meeting it in point Q.
The original triangle APB is now split into two triangles, APQ and BPQ which I shall prove are CONGRUENT (in that order of vertices):
(i) Triangles APQ and BPQ contain right angles at Q. (ii) As given, lengths AP and BP are equal; by construction, they are the HYPOTENEUSES of the two right-angled triangles. (iii) The length PQ is in common.
Therefore the triangles are congruent (by RHS) [The latter mnemonic stands for "Right angle, Hypoteneuse, Side," one of the standard ways of proving congruence.]
Hence AQ = BQ (corresponding sides in two congruent triangles), and the perpendicular that was dropped from P onto AB is therefore a PERPENDICULAR BISECTOR, so that P is on the perpendicular bisector.
QED
[Incidentally, this is how you would also PROVE that the angles A and B are equal (or "congruent" using the faddish modern equivalent) in an isosceles triangle with sides AP = PB. It would be engaging in a circular argument to attempt to use that "fact" to prove the bisection property.]
Live long and prosper.
2007-08-31 09:26:38 · answer #1 · answered by Dr Spock 6 · 1⤊ 0⤋
I think the question you are trying to ask is a geometric proof that uses a compass, a straight edge, and a pencil as your only tools. The compass is not the navigation type, it is the one that has a sharp point on one end and a pencil in the other end and works quite well for drawing circles. The question I think you are trying to ask is to draw a line segment, then draw a line bisects that segment pependicularly. Draw a segment with your straight edge and pencil. Then adjust the compass so its a little over half the length of the segment. Then put the tip of the compass on one end of the segment and draw an arc above and below the line, in line with where the center of the segment looks like it will be. Then put the tip of the compass on the other end of the segment and draw an arc above and below the segment. Then use your straight edge and pencil and draw a line from where the top two arcs intersect, thru the segment to where the bottom two arcs intersect. Congratulation, you have just drawn a segments perpendicular bisector.
2007-08-31 09:26:07 · answer #2 · answered by mythoughts 2 · 0⤊ 1⤋
If you draw it out you get a triangle with two congruent sides and two congruent angles.
Drawing a perpendicular line to segment should set up a picture that allows you to prove that the line is a bisector as well...
2007-08-31 09:28:25 · answer #3 · answered by Bear 2 · 0⤊ 1⤋
The perpendicular bisector intersects a segment at its midpoint. If the point P is equidistant from the two endpoints then it *must* be the midpoint, meaning that it also lies on the bisector.
2007-08-31 09:27:23 · answer #4 · answered by Mathsorcerer 7 · 0⤊ 2⤋
how can a point lie on "the two ends". I dont get it at all!
2007-08-31 09:20:10 · answer #5 · answered by Mike M 4 · 0⤊ 0⤋
find the slope of the two ends and take the negative multiplicative of the slope
2007-08-31 09:23:12 · answer #6 · answered by Anonymous · 0⤊ 2⤋
OH you hurt cheng's brain.
2016-05-18 00:10:55 · answer #7 · answered by shannan 3 · 0⤊ 0⤋