1984 AIME Problems/Problem 3
Contents
Problem
A point is chosen in the interior of
such that when lines are drawn through
parallel to the sides of
, the resulting smaller triangles
,
, and
in the figure, have areas
,
, and
, respectively. Find the area of
.
![[asy] size(200); pathpen=black;pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE); [/asy]](http://latex.artofproblemsolving.com/2/9/3/293009bb673bfe7d5b163c98428082f101000e4d.png)
Solution 1
By the transversals that go through , all four triangles are similar to each other by the
postulate. Also, note that the length of any one side of the larger triangle is equal to the sum of the sides of each of the corresponding sides on the smaller triangles. We use the identity
to show that the areas are proportional (the sides are proportional and the angles are equal) Hence, we can write the lengths of corresponding sides of the triangle as
. Thus, the corresponding side on the large triangle is
, and the area of the triangle is
.
Solution 2
Alternatively, since the triangles are congruent by , then the ratios between the bases and the heights of each of the three triangles would all be equal. The areas of each of the triangles are all perfect squares, so we could assume
That means that the base of
is 4, the base of
is 6, and the base of
is 14. Since the quadrilaterals underneath
and
are both parallelograms, and opposite sides of a parallelogram are congruent, the base of the large triangle is
. Therefore, the height of the entire triangle would be twelve, so therefore, the area of the large triangle is
.
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |