Animated Solution for Mathematics - Straight Lines: The vertices of a triangle are $A(-1,-7),B(5,1)$ and $C(1,4)$. The equation of the bisector of the angle $\angle ABC$ is .........

Unveiling the Angle Bisector

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are exploring the elegant dance between geometry and algebra.

We have a triangle, $△ABC$, with vertices $A(-1,-7)$, $B(5,1)$, and $C(1,4)$. Our mission is to find the equation of the internal bisector of $\angle ABC$.

This is a classic problem that tests your ability to bridge the gap between visual intuition and rigorous calculation.

Visualizing the Canvas

Before we touch a single equation, let's ground ourselves. Imagine the coordinate plane with three points, $A$, $B$, and $C$.

The angle bisector of $\angle ABC$ is a line that originates at vertex $B$ and slices through the triangle, splitting the angle at $B$ into two perfectly equal parts. This line eventually intersects the opposite side, $AC$, at a point we shall call $D$.

Our goal is to find the equation of the line $BD$. To do this, we need the coordinates of $B$ (which we have) and the coordinates of $D$.

The Angle Bisector Theorem

This is where the magic happens. We invoke the Angle Bisector Theorem.

This theorem is a cornerstone of Euclidean geometry, stating that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle. Mathematically, this means:

This is our master plan. If we can find the lengths of $AB$ and $BC$, we know exactly where $D$ sits on the segment $AC$.

The Calculation

Let's calculate the lengths. Using the distance formula, $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, we find:

For $AB$:

For $BC$:

Look at that! The ratio $\frac{AB}{BC}=\frac{10}{5}=\frac{2}{1}$. This tells us that point $D$ divides the segment $AC$ in a $2:1$ ratio.

The Section Formula

Now, we use the section formula to find the coordinates of $D$. If a point divides the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, the coordinates are:

Here, $m=2$, $n=1$, $A=(-1,-7)$, and $C=(1,4)$.

So, $D=\left(\frac{1}{3},\frac{1}{3}\right)$.

The Final Line

We have two points on our line: $B(5,1)$ and $D\left(\frac{1}{3},\frac{1}{3}\right)$. The slope $m$ of the line $BD$ is:

Using the point-slope form, $y-y_1=m(x-x_1)$, we get:

And there it is! The equation of the angle bisector is $x-7y+2=0$. By combining the geometric insight of the Angle Bisector Theorem with the algebraic power of the section formula, we have conquered the problem.