Angle Bisectors


I’ll cover one more topic before bringing the straight line series to a (temporary) close – Angle Bisectors.

That is, to find the equation to the lines, which bisect the angle between two given lines.

There are again two different methods using which we can obtain their equations. One is quite lengthy, and the other one is quite elegant. I’ll begin with the boring one first.

Method 1

Consider two given lines a1x + b1y + c= 0 and a2x + b2y + c= 0. We have to find the equation of the lines which pass through their point of intersection, and make equal angles with both these lines.

We’ll start first by obtaining the point of intersection of the lines (here’s how). Let’s say the point obtained is (x1, y1).

Now we need to find the slope of the lines. Let the required slope be ‘m’. Then, using the expressions for angle between two lines, we’ll equate the angle between the required line and the given two lines.

That is, \( \left | \frac{m-m_1}{1+mm_1}  \right | = \left | \frac{m-m_2}{1+mm_2}  \right | \)

Solving this equation will give us two different values of m (because of the modulus), which means two different lines, that is, two different angle bisectors.

straight line angle bisectors coordinate geometry

Now this method is quite a bit lengthy, as it involves finding the intersection point, followed by solving equations to find the slopes.

This inspires me to find another method, which is coming up next !

Method 2

This one is pretty straightforward. Consider the two lines a1x+b1y+c1=0 and a2x+b2y+c2=0 again.

Now, any point on the angle bisector will be equidistant from the two lines.

straight line angle bisectors coordinate geometry

If P(x, y) be any point on the angle bisector, then using this formula, we can write \( \frac{|a_1x+b_1y+c_1|}{\sqrt{{a_1}^2 + {b_1}^2}}=\frac{|a_2x+b_2y+c_2|}{\sqrt{{a_2}^2 + {b_2}^2}}\).

And that’s pretty much it. This is the required equation!

On removing the modulus, we get two equations, as we’ll have two bisectors of the angles formed between the lines: \(\frac{a_1x+b_1y+c_1}{\sqrt{{a_1}^2 + {b_1}^2}}=\pm\frac{a_2x+b_2y+c_2}{\sqrt{{a_2}^2 + {b_2}^2}}\)

Lesson Summary

  1. The equation of the angle bisectors to the pair of straight lines a1x+b1y+c1=0 and a2x+b2y+c2=0 will be given by \( \frac{a_1x+b_1y+c_1}{\sqrt{{a_1}^2 + {b_1}^2}}=\pm\frac{a_2x+b_2y+c_2}{\sqrt{{a_2}^2 + {b_2}^2}}\)

That’s it for this one. Next, I’ll cover some examples related to angle bisectors. See you soon !

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