+128 In $\triangle ABC$, shown here, $\overline{AB}$ and $\overline{AC}$ have each been divided into four congruent segments. What fraction of triangle $ABC$ is shaded? Express your answer as a common fraction.
https://latex.artofproblemsolving.com/7/3/4/734d6369be3ad3401903d79373fba68f790af690.png
+128460 Let the area of triangle ABC = A
Discounting the bottom unshaded area the remaining triangle is (3/4)^2 * area of ABC =
(9/16) A
So....the area of the bottom unshaded area = A - (9/16)A = (7/16)A (1)
Looking at the "top" triangle composed of the top shaded area and the underneath unshaded area......the area of this triangle = (1/2)^2 * area of ABC = (1/4)A
And the area of the top shaded area = (1/4)^2 * area of ABC = (1/16)A
So....the area of the second unshaded region = (1/4)A - (1/16)A = (3/16)A (2)
So.....the area of the shaded regions =
area of ABC - the areas of the two unshaded regions =
area of ABC - (1) - (2) =
A - (7/16)A - (3/16)A =
(6/16)A = (3/8) A
So....the shaded areas are (3/8) the area of triangle ABC
