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

Place \(A(0,0),\ B(w,0),\ C(w,h),\ D(0,h)\). Then \[ P=\left(\tfrac{w}{2},\,h\right),\qquad Q=\left(w,\,\tfrac{h}{2}\right). \] Unshaded region is \(\triangle APQ\). Its area \[ [APQ]=\tfrac12\left|\det\!\begin{pmatrix}\tfrac{w}{2}& h \\[2pt] w & \tfrac{h}{2}\end{pmatrix}\right| =\tfrac12\left|\tfrac{wh}{4}-wh\right| =\tfrac{3wh}{8}. \] Rectangle area \(=wh\), so shaded area \(=wh-\tfrac{3wh}{8}=\tfrac{5wh}{8}\). Hence \[ \text{Shaded:Unshaded}=\frac{5/8}{3/8}= \boxed{5:3}. \]

Answer 2

Instead of computing the area of triangle \(APQ\) with a determinant formula, we can find the shaded area directly as the sum of the three corner triangles left over outside \(\triangle APQ\), and check each option.

1. Option A (\(5:4\)): Place \(A(0,0)\), \(B(w,0)\), \(C(w,h)\), \(D(0,h)\), so \(P=\left(\tfrac{w}{2},h\right)\) and \(Q=\left(w,\tfrac{h}{2}\right)\). The shaded region (rectangle minus \(\triangle APQ\)) splits into three right triangles: \(\triangle ADP\) with legs \(AD=h\) and \(DP=\tfrac w2\), area \(\tfrac{wh}{4}\); \(\triangle PCQ\) with legs \(PC=\tfrac w2\) and \(CQ=\tfrac h2\), area \(\tfrac{wh}{8}\); and \(\triangle QBA\) with legs \(QB=\tfrac h2\) and \(BA=w\), area \(\tfrac{wh}{4}\). Their sum is \(\tfrac{wh}{4}+\tfrac{wh}{8}+\tfrac{wh}{4}=\tfrac{5wh}{8}\), which is the shaded area; the unshaded \(\triangle APQ\) is the remaining \(wh-\tfrac{5wh}{8}=\tfrac{3wh}{8}\). This gives a ratio of \(5:3\), not \(5:4\), ruling out this option.
2. Option B (\(3:5\)): This is the inverse of the correct ratio \(5:3\), so it is incorrect.
3. Option C (\(5:3\)): As computed, shaded\(:\)unshaded \(=\tfrac{5wh}{8}:\tfrac{3wh}{8}=5:3\), matching this option.
4. Option D (\(5:8\)): This compares the shaded area to the whole rectangle rather than to the unshaded area, so it does not answer the question asked.

Summing the three corner triangles confirms the shaded-to-unshaded ratio is \(5:3\).

Hence, the correct answer is option C: \(5:3\).