Question

In the given figure, \(PQRS\) is a parallelogram with \(PS = 7\) cm, \(PT = 4\) cm and \(PV = 5\) cm. What is the length of \(RS\) in cm? (The diagram is representative.) \begin{center} \includegraphics[width=0.5\textwidth]{01.jpeg} \end{center}

• A. \(\dfrac{20}{7}\)
• B. \(\dfrac{28}{5}\)
• C. \(\dfrac{9}{2}\)
• D. \(\dfrac{35}{4}\)

Hint:

For any parallelogram, area \(=\) (any side) \(\times\) (perpendicular height to that side). If two different heights are given to two different sides, equate the two area forms.

Answer 1

The Correct Option is B

Step 1: Interpret the perpendiculars.
From the figure, \(PT \perp QR\). Since \(PS \parallel QR\), the perpendicular distance (height) between the parallel lines \(PS\) and \(QR\) equals \(PT=4\) cm.
Also, \(PV \perp RS\), so \(PV\) is the height to side \(RS\).

Step 2: Use equal-area expressions of a parallelogram.
Area of \(PQRS\) using base \(PS\): \(A = (\text{base})\times(\text{height}) = 7 \times 4 = 28\) \(\text{cm}^2\).
Area using base \(RS\): \(A = RS \times PV = RS \times 5\).

Step 3: Equate the two areas and solve for \(RS\).
\[ RS \times 5 = 28 \;\Rightarrow\; RS = \frac{28}{5}\ \text{cm}. \]

Final Answer:
\[ \boxed{\dfrac{28}{5}\ \text{cm}} \]