Distinguishing between angular bisectors

In summary, if the two lines have a slope that is positive, then the bisector that bisects the external angle is the bisector that you are looking for. If the slope is negative, then the bisector that bisects the internal angle is the bisector that you are looking for.
  • #1
JC2000
186
16

Homework Statement

:[/B]
The following expression stands for the two angular bisectors for two lines :[tex]\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad[/tex]

Homework Equations


The equations for the two lines are :
##a_1x + b_1y + c_1 = 0~##and ##a_2x + b_2y + c_2 = 0~##

The Attempt at a Solution


One way is to find the slope from the equation for the respective bisectors and then use the slope of one of the given lines and then apply the formula for angle between two lines to check if the bisector is acute or not. I was wondering if there was a more direct way to do this.

My questions :
(A)
Since the two lines can be bisected in two ways, is there a formula to distinguish between the bisector that bisects the external angle and the bisector that bisects the internal or acute angle between the two lines?
My book says that if [tex] a_1 * a_2 + b_1 * b_2 > 0[/tex] then this refers to the external bisector.
I am stumped by this because : (B) How is this expression arrived at? (C) Since [tex]a_1, a_2, b_1, b_2[/tex] all refer to the original equations of the two lines and not the bisectors, which bisector do this expression refer to?
 
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  • #2
JC2000 said:

Homework Statement

:[/B]
The following expression stands for the two angular bisectors for two lines :[tex]\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad[/tex]

Homework Equations


The equations for the two lines are :
[tex]ax_1 + by_1 + c_1 = 0[/tex]
and [tex]ax_2 + by_2 + c_2 = 0[/tex]

The Attempt at a Solution


One way is to find the slope from the equation for the respective bisectors and then use the slope of one of the given lines and then apply the formula for angle between two lines to check if the bisector is acute or not. I was wondering if there was a more direct way to do this.

My questions :
(A)
Since the two lines can be bisected in two ways, is there a formula to distinguish between the bisector that bisects the external angle and the bisector that bisects the internal or acute angle between the two lines?
My book says that if [tex] a_1 * a_2 + b_1 * b_2 > 0[/tex] then this refers to the external bisector.
I am stumped by this because : (B) How is this expression arrived at? (C) Since [tex]a_1, a_2, b_1, b_2[/tex] all refer to the original equations of the two lines and not the bisectors, which bisector do this expression refer to?
Did you mean to write the equations of the two lines as
##a_1x + b_1y + c_1 = 0~##and ##a_2x + b_2y + c_2 = 0~##?
 
  • #3
kuruman said:
Did you mean to write the equations of the two lines as
##a_1x + b_1y + c_1 = 0~##and ##a_2x + b_2y + c_2 = 0~##?

Yes. I will edit the formatting now...Thanks!
 
  • #4
It turns out that you can find the answer to this problem on several websites with several different approaches. I recommend that you do the research.
 

Related to Distinguishing between angular bisectors

1. What is an angular bisector?

An angular bisector is a line that divides an angle into two equal parts. It passes through the vertex of the angle and splits it into two congruent angles.

2. How do you distinguish between angular bisectors and regular bisectors?

Regular bisectors divide a line segment into two equal parts, while angular bisectors divide an angle into two equal parts.

3. How can you find the angular bisector of an angle?

To find the angular bisector of an angle, you can use a compass and a straightedge. Place the compass at the vertex of the angle and draw two arcs that intersect the two sides of the angle. Then, draw a line through the vertex and the point where the two arcs intersect.

4. What is the importance of angular bisectors?

Angular bisectors are important in geometry and trigonometry as they help in solving problems involving angles and triangles. They also play a crucial role in constructions and proofs.

5. Can an angle have more than one angular bisector?

No, an angle can only have one angular bisector. This is because the bisector must pass through the vertex and divide the angle into two equal parts. If there were more than one bisector, the angle would be divided into more than two equal parts, which is not possible.

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