Question

A triangle has vertices $(0, 0),$ $(1, 1),$ and $(6m, 0),$ and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m?$

$-\dfrac{1}{3}$

$-\dfrac{1}{6}$

$\dfrac{1}{6}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Solution:

The line $y = mx$ passes through the vertex $(0, 0),$ so it bisects the triangle's area exactly when it passes through the midpoint of the opposite side, joining $(1, 1)$ and $(6m, 0).$ That midpoint is $\left(\dfrac{6m + 1}{2}, \dfrac{1}{2}\right).$

Requiring it to satisfy $y = mx$ gives $\frac{1}{2} = m\cdot\frac{6m + 1}{2},$ so $6m^2 + m - 1 = 0,$ that is $(3m - 1)(2m + 1) = 0.$

The possible values are $m = \dfrac{1}{3}$ and $m = -\dfrac{1}{2},$ whose sum is $-\dfrac{1}{6}.$

Thus, the correct answer is B.

2009 AMC 12A Problem 14

Solution:

Problem 14 in Other Years