Line Splits Triangle Into Equal Areas

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

Show/Hide Solution

The area of △OBD is split in half by line CA. △OBD has base BD = 8 and altitude EO = 1. So area △OBD = (8 x 1)/2 = 4. This means area △ABC = 2 = a(CA)/2.

We know CA||OE, so △BAC ∼ △BOE, and CA/OE = BC/BE, i.e., CA/1 = a/9.

Now area △ABC = a(CA)/2 = (a)(a)/(2)(9) = a²/18 and this area is 2. But a²/18 = 2 → a = 6.

Thus line CA intersects the x-axis at x=3, and that’s the equation of line CA: x = 3.

The answer is (b).