Does Bisecting Angle A in a 3-4-5 Triangle Divide It into Two Equal Areas?

[Aceterp]

Given triangle ABC. And given that it’s a common 3-4-5 right triangle. (this still qualifies as a scalene)

So

AB = 4

BC = 3

AC = 5 is the hypotenuse

If Angle A was bisected, do the two newly formed triangles have the same area? And would the bisecting line hit the midpoint of BC?

Is there a triangle law that applies to this question? My inital thought is that the bisecting angle line doen't not necessaily cross the midpoint of BC. looking for a proof or any help.

Thanks!

 

[Mark44]

Given triangle ABC. And given that it’s a common 3-4-5 right triangle. (this still qualifies as a scalene)

So

AB = 4

BC = 3

AC = 5 is the hypotenuse

If Angle A was bisected, do the two newly formed triangles have the same area? And would the bisecting line hit the midpoint of BC?

Is there a triangle law that applies to this question? My inital thought is that the bisecting angle line doen't not necessaily cross the midpoint of BC. looking for a proof or any help.

Thanks!

Angle A = arctan(3/4), so the bisected angle is half of that. Let D be the point where the angle bisector hits BC. Since you know angle DAB, angle BDA is the complement of angle DAB. Now you know all three angles (angle ABC is a right angle), and one side (AB = 4), so you can use the Law of Sines to find the other side, BD. Hopefully you can take it from there.