Question 6

The diagram shows a rectangle $$ABCD$$ where $$AB\colon AD=1\colon2$$. Point $$E$$ on $$AC$$ is such that $$DE$$ is perpendicular to $$AC$$. What is the ratio of the area of the triangle $$DCE$$ to the rectangle $$ABCD$$?

Solution

Let $$AB=x$$ and $$AD=2x$$. Using coordinates with $$A=(0,0)$$, $$C=(x,2x)$$ and $$D=(0,2x)$$, the foot of the perpendicular from $$D$$ to $$AC$$ is $$E=(4x/5,8x/5)$$. Thus the height of triangle $$DCE$$ is $$2x/5$$ and its base is $$x$$, giving area $$x^2/5$$. The rectangle has area $$2x^2$$, so the required ratio is $$1\colon10$$.

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The diagram shows a rectangle $$ABCD$$ where $$AB\colon AD=1\colon2$$. Point $$E$$ on $$AC$$ is such that $$DE$$ is perpendicular to $$AC$$. What is the ratio of the area of the triangle $$DCE$$ to the rectangle $$ABCD$$?

Solution

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