Animated Solution for Mathematics - Straight Lines: Let $P=(-1,0),Q=(0,0)$ and $R=(3,3\sqrt{3})$ be three points. The equation of the bisector of the angle $PQR$ is

When you look at the points $P(-1,0)$, $Q(0,0)$, and $R(3,3\sqrt{3})$, do not just see numbers. See a map.

Point $Q$ is our anchor at the origin. Point $P$ is a sentinel on the negative $x$-axis, and $R$ is a beacon in the first quadrant. Our mission is to find the line that cuts the angle $\angle PQR$ perfectly in half.

The Angular Map

Before we touch an equation, we must visualize. Ray $QR$ connects the origin to $(3,3\sqrt{3})$.

To find its inclination, we calculate the slope:

We know that $\tan (\theta )=\sqrt{3}$, which tells us that $\theta =60^{\circ }$. This is our first ray.

Now, look at ray $QP$. It lies on the negative $x$-axis. In the language of trigonometry, the angle measured from the positive $x$-axis to the negative $x$-axis is exactly $180^{\circ }$.

We have our two boundaries: $60^{\circ }$ and $180^{\circ }$.

The Art of Bisection

The total angle $\angle PQR$ is the difference between these two inclinations: $180^{\circ }-60^{\circ }=120^{\circ }$. This is an obtuse angle, and we need to slice it right down the middle.

The bisector will be a line that splits this $120^{\circ }$ into two $60^{\circ }$ segments. Starting from our ray $QR$ at $60^{\circ }$, we add half of the total angle:

This $120^{\circ }$ is the inclination of our bisector line.

The Final Equation

Now, we bring it home. We have a line passing through the origin $(0,0)$ with an inclination of $120^{\circ }$.

The slope $m$ is $\tan (120^{\circ })$. Using the identity $\tan (180^{\circ }-60^{\circ })=-\tan (60^{\circ })$, we find:

The equation of a line through the origin is $y=mx$. Substituting our slope, we get $y=-\sqrt{3}x$.

Rearranging this into the standard form, we arrive at the final result:

Look at that elegance! The math didn't just give us an answer; it revealed the perfect balance of the system. Keep practicing this visualization, and you will find that geometry becomes less about memorizing formulas and more about seeing the truth of the shapes.