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jmc

geometry intermediate

Problem

Triangle is a right isosceles triangle. Points , and are the midpoints of the sides of the triangle. Point is the midpoint of segment and point is the midpoint of segment . What is the ratio of the shaded area to the non-shaded area in triangle ? Express your answer as a common fraction.

problem
Solution
and by the midline theorem and is right, so is a rectangle. , so and is a square. Say it has side length ; and . has area , has area , and has area . The shaded area is thus , and the non-shaded area is . Therefore, the ratio of shaded to nonshaded area is
Final answer
\frac{5}{11}