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).

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