+0

# Coordinates

-1
23
1
+1475

The line $y = mx$ bisects the angle between the two lines shown below. Find $m$.

Feb 25, 2024

#1
+398
+3

We have a very interesting property, and very useful property when it comes to slope: $$\tan=\frac{O}{A}$$, opposite over adjacent, but when put in the coordinate plane, that is $$\tan=\frac{rise}{run}$$, and what else is rise over run? Slope! .

Set the angle formed by y=5x and the x axis as $$\theta$$, and the angle formed by y=1/6x and the x axis as $$\phi$$.

We want to find $$\tan(\phi+\frac{\theta-\phi}{2})=\tan(\frac{\theta+\phi}{2})$$.

Use some of our identities:

With a lot of use of tangent identities, we know $$\tan(\theta)=5, \tan(\phi)=1/6$$.

Directly applying our tangent angle identity: $$\tan(\theta+\phi)=\frac{5+\frac{1}{6}}{1-\frac{5}{6}}=31$$.

Calculate, $$\tan(\theta+\phi)=\frac{\sin(\theta+\phi)}{1+\cos(\theta+\phi)}$$.

Draw a triangle and calculate sin and cos.

$$\sin(\theta+\phi)=\frac{31}{\sqrt{962}}$$$$\cos(\theta+\phi)=\frac{1}{\sqrt{962}}$$

Substituting in, we get $$y=(\frac{\sqrt{962}-1}{31})x$$ as our line, so $$\bf{m=\frac{\sqrt{962}-1}{31}}$$.

Feb 27, 2024