Question

AB, CD and EF are three parallel lines, in that order. Let $d_1$ and $d_2$ be the distances from CD to AB and EF respectively. $d_1$ and $d_2$ are integers, where $d_1:d_2=2:1$. $P$ is a point on AB, $Q$ and $S$ are points on CD and $R$ is a point on EF. If the area of the quadrilateral $PQRS$ is $30$ square units, what is the value of $QR$ when the value of $SR$ is the least?

• A. slightly less than $10$ units
• B. $10$ units
• C. slightly greater than $10$ units
• D. slightly less than $20$ units
• E. slightly greater than $20$ units

Hint:

With parallel lines, area between them equals $\tfrac12 \times$ (distance between the lines) $\times$ (base on one line). Use this to express unknown segments, and for “minimum distance” between parallels, choose the perpendicular.

Answer 1

Answer: (E)

Step 1: Express the area in terms of $d_1,d_2$ and $QS$.
Since AB, CD, EF are parallel, $PQRS$ splits into two triangles with the same base $QS$: $\triangle PQS$ (between AB and CD) and $\triangle QRS$ (between CD and EF). Hence \[ [ PQRS ] \;=\; \tfrac12 d_1\cdot QS \;+\; \tfrac12 d_2\cdot QS \;=\; \tfrac12 (d_1+d_2)\,QS. \] Given $d_1:d_2=2:1$, let $d_1=2k,\; d_2=k$ with $k\in\mathbb{Z}^+$. With area $30$, \[ 30=\tfrac12(3k)\,QS \;\Rightarrow\; QS=\frac{20}{k}. \tag{1} \]

Step 2: Minimizing $SR$.
For fixed parallel lines, the shortest segment from CD to EF is the perpendicular distance $d_2=k$. The minimum of $SR$ occurs when $SR\perp$ (CD, EF), i.e., when $R$ is directly above $S$; thus \[ SR_{\min}=k. \tag{2} \]

Step 3: Find $QR$ when $SR$ is minimum.
With $R$ vertically above $S$, $QR$ is the hypotenuse of a right triangle with legs $QS$ (along CD) and $SR_{\min}=k$: \[ QR=\sqrt{(QS)^2+k^2}=\sqrt{\left(\frac{20}{k}\right)^2+k^2}. \tag{3} \] To keep $SR$ as small as possible we choose the smallest integer $k$, namely $k=1$ (since both $d_1=2k$ and $d_2=k$ must be integers). Then from (1)–(3): \[ QS=20,\quad SR_{\min}=1,\quad QR=\sqrt{20^2+1^2}=\sqrt{401}\approx 20.0249\ldots \] which is \emph{slightly greater than $20$}.

Final Answer: \[ \boxed{\text{(E) slightly greater than 20 units}} \]