I have tried using sine theoram, angle bisector theoram, congruency of type RHS,AA,ASA but haven't been able to do this.

Aidyn Crosby

Aidyn Crosby

Answered question


How to prove that angle bisector of right angle triangle ABC right angled at B is perpendicular bisector of third side AC

Answer & Explanation



Beginner2022-09-07Added 10 answers

Another proof that this is only true for a 45-45-90 right triangle.
If the angle bisector of the right angle is perpendicular to the hypotenuse, then the triangle formed by the bisector has a 45 angle (bisects the right angle) and a 90 angle (perpendicular) so the other angle has to be 45.
Therefore the original triangle is 45-45-90.


Beginner2022-09-08Added 1 answers

I don't think the statement you are trying to prove is true for any right triangle. Here is why.
Let the two adjacent sides of the right triangle be represented by vectors, A = (0,a), B = (b,0), then the hypotenuse of the triangle would be B-A = (b,-a) by the parallelogram law. Let the vector V = (1,1)(in fact, any vector with angle 45 degrees). Note that V is an angle bisector of the right angle.
Now, if we take the dot product of V and B-A, and set it equal to 0(since we want them to be perpendicular). We get b-a = 0. Thus V is perpendicular to the other side if and only if b = a.

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