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

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In summary, the question is whether two newly formed triangles, after bisecting angle A of a 3-4-5 right triangle, have the same area and if the bisecting line hits the midpoint of BC. There is a triangle law, the Law of Sines, that can be used to find the other side of the triangle and potentially prove that the bisecting line hits the midpoint of BC.
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Aceterp
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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!
 
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  • #2
Aceterp said:
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.
 

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

1. What is the "Triangle Bisection question"?

The Triangle Bisection question is a mathematical problem that involves finding the point of intersection of a line segment drawn from one vertex of a triangle to the opposite side, which divides the segment into two equal parts.

2. What is the significance of the Triangle Bisection question?

The Triangle Bisection question has applications in geometry, trigonometry, and engineering. It is used to solve various problems involving triangles, such as finding the center of gravity or the centroid of a triangle.

3. How is the Triangle Bisection question solved?

The Triangle Bisection question can be solved using various methods, including the midpoint formula, the properties of similar triangles, or the Pythagorean theorem. The specific method used depends on the given information and the desired result.

4. Can the Triangle Bisection question be applied to all types of triangles?

Yes, the Triangle Bisection question can be applied to all types of triangles, including equilateral, isosceles, and scalene triangles. However, the method of solving may differ based on the type of triangle.

5. Are there any real-world applications of the Triangle Bisection question?

Yes, the Triangle Bisection question has real-world applications in various fields such as architecture, surveying, and navigation. It is used to determine the location of objects or structures in relation to a given point or area.

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