Difference between revisions of "1984 AIME Problems/Problem 3"

Problem

A point $\displaystyle P$ is chosen in the interior of $\displaystyle \triangle ABC$ such that when lines are drawn through $\displaystyle P$ parallel to the sides of $\displaystyle \triangle ABC$, the resulting smaller triangles $\displaystyle t_{1}$, $\displaystyle t_{2}$, and $\displaystyle t_{3}$ in the figure, have areas $\displaystyle 4$, $\displaystyle 9$, and $\displaystyle 49$, respectively. Find the area of $\displaystyle \triangle ABC$.

Solution

Notice that all four triangles are similar to each other. Also, note that the length of any one side of the larger triangle is equation to the sum of the sides of each of the corresponding sides on the smaller triangles. Since the squares of the lengths of the sides are in proportion with the area, we can write the lengths of corresponding sides of the triangle as $2x,\ 3x,\ 7x$. Thus, the corresponding side on the large triangle is $12x$, and the area of the triangle is $12^2 = 144$.