Question

In the given figure, $ABCD$ is a rectangle. $P$ and $Q$ are the midpoints of sides $CD$ and $BC$ respectively. Then the ratio of area of shaded portion to the area of unshaded portion is:

• A. $5:4$
• B. $3:5$
• C. $5:3$
• D. $5:8$

Hint:

Set convenient coordinates for geometry-in-rectangles. Midpoints give clean fractions; use the $2$D determinant for triangle areas.

Answer 1

The Correct Option is C

With \(A(0,0)\), \(B(w,0)\), \(C(w,h)\), \(D(0,h)\), \(P=\left(\tfrac w2,h\right)\), \(Q=\left(w,\tfrac h2\right)\), the shaded region is made of three corner right triangles:

1. \(\triangle ADP\): legs \(h,\tfrac w2\), area \(\tfrac{wh}{4}\)
2. \(\triangle PCQ\): legs \(\tfrac w2,\tfrac h2\), area \(\tfrac{wh}{8}\)
3. \(\triangle QBA\): legs \(\tfrac h2,w\), area \(\tfrac{wh}{4}\)

Sum \(=\tfrac{5wh}{8}\) (shaded); unshaded \(\triangle APQ=wh-\tfrac{5wh}{8}=\tfrac{3wh}{8}\). Ratio shaded:unshaded \(=5:3\).