# 3460.11 – Split That Triangle!

A vertical line divides the triangle with vertices $(0,0), (1,1)$ and $(9,1)$ in the $xy$-plane into two regions of equal area. What is the equation of the line?

1. $x = 2.5$
2. $x = 3.0$
3. $x = 3.5$
4. $x = 4.0$
5. $x = 4.5$

Solution

Please refer to the diagram below.

The area of $\Delta\mathsf{OBD}$ is split in half by line $\mathsf{CA}$. Since $\Delta\mathsf{OBD}$ has base $\mathsf{BD} = 8$ and altitude $\mathsf{EO} = 1$, its area is

$\mathcal{A}(\triangle \mathsf{OBD}) = (8 \times 1)/2 = 4.$

This means

$\mathcal{A}(\triangle \mathsf{ABC}) = 2 = a(\mathsf{CA})/2.$

Now $\mathsf{CA} \parallel \mathsf{OE}$, so triangles $\Delta\mathsf{BAC}$ and $\Delta\mathsf{BOE}$ are similar and $\mathsf{CA/OE = BC/BE}$, i.e., $\mathsf{CA/1 = a/9}$. Now,

$\mathcal{A}(\triangle \mathsf{ABC}) = a(CA)/2 = a^2/18.$

But this area is also 2. Therefore, $a^2/18 = 2$ and so $a = 6$.

Thus line $\mathsf{CA}$ intersects the $x$-axis at $x=3$, and that's the equation of line $\mathsf{CA}$$x = 3$.