Question

Prove that bisector of a triangle passes through the same point

Answer 1

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CONSTRUCTION: In triangle ABC , OQ & OR the perpendicular bisectors of sides BC & AC respectively, intersecting at point O. P is the mid point of AB.

TO PROVE THAT: OP is also a perpendicular bisector of AB. In that case , we can say that all 3 perpendicular bisectors are concurrent.

PROOF:

Since any point on the perpendicular bisector of any segment is equidistant from the end points of the segment. ( theorem proved by SAS congruency) . And also the converse of this theorem is true & can be proved by SAS congruency)

=> OB = OC

And also OC = OA

So, by above 2 statements, OB = OA. ………(1)

Also, PB = PA ( as P is the mid point of AB , by construction)

Hence, O & P both lie on the perpendicular bisector of AB. ( as these points are equidistant from A & B)

=> OP is the third perpendicular bisector of the triangle.

=> All 3 perpendicular bisectors have common point O.

Hence, these 3 are concurrent

[ Hence Proved]

Answer 2

Answer:

In a triangle, bisector is the line which divides a side of the triangle into two equal halves. In other words, the bisector will always intersect at the mid points of a side of the triangle. In a triangle, perpendicular bisector is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

I Think that this figure will help you.