Question

In the given figure, \(PQRS\) is a parallelogram with \(PS = 7\text{ cm}\), \(PT = 4\text{ cm}\) and \(PV = 5\text{ cm}\). 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 parallelograms, the area is invariant across any base–height pair. If two altitudes are given to two different sides, set \( (\text{base}_1)(\text{height}_1) = (\text{base}_2)(\text{height}_2) \) and solve.

Answer 1

The Correct Option is B

Step 1: Identify parallel sides and altitudes.
In a parallelogram, opposite sides are parallel: \(PS \parallel QR\) and \(PQ \parallel RS\).
From the figure, \(PT \perp QR\) with \(PT=4\), and \(PV \perp RS\) with \(PV=5\).

Step 2: Compute area using base \(PS\).
Because \(PS \parallel QR\) and \(P \in PS\), the perpendicular distance from \(PS\) to \(QR\) equals \(PT\).
\[ \text{Area} = (\text{base } PS)\times(\text{height to } PS)=PS \times PT = 7 \times 4 = 28 \text{ cm}^2. \]

Step 3: Equate the area using base \(RS\).
Using base \(RS\) and its corresponding altitude \(PV\):
\[ \text{Area} = RS \times PV \;\Rightarrow\; 28 = RS \times 5 \;\Rightarrow\; RS = \dfrac{28}{5}\text{ cm}. \]

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