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.) 

 

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

Hint:

In geometry problems, if multiple lengths and heights are given for a single shape, it's a strong hint that you should calculate a property (like area or volume) in more than one way and then equate the expressions to find an unknown variable.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question requires using the properties of a parallelogram, specifically the formula for its area. The area of a parallelogram can be calculated using any side as the base and the corresponding perpendicular height.
Step 2: Key Formula or Approach:
The area of a parallelogram is given by the formula: \[ \text{Area} = \text{base} \times \text{height} \] A key property of a parallelogram is that opposite sides are equal in length. Therefore, \(PQ = RS\) and \(PS = QR\).
Step 3: Detailed Calculation:
The area of the parallelogram PQRS can be calculated in two ways using the given information.
Method 1: Using base QR and height PT
- The length of side PS is given as 7 cm.
- Since PQRS is a parallelogram, the length of the opposite side QR is equal to PS. So, \(QR = PS = 7\) cm.
- The height corresponding to the base QR is PT, which is given as 4 cm.
- Area of PQRS = \(QR \times PT = 7 \times 4 = 28\) cm\(^2\).
Method 2: Using base RS and height PV
- We need to find the length of side RS.
- The height corresponding to the base RS is PV, which is given as 5 cm.
- Area of PQRS = \(RS \times PV = RS \times 5\).
Equating the two areas:
Since the area of the parallelogram is the same regardless of which base and height are used, we can equate the two expressions for the area.
\[ RS \times 5 = 28 \] \[ RS = \frac{28}{5} \] Step 4: Final Answer:
The length of RS is \(\frac{28}{5}\) cm.
Step 5: Why This is Correct:
The solution correctly applies the formula for the area of a parallelogram and the property that opposite sides are equal. By calculating the area in two different ways and equating the results, we can solve for the unknown side length RS. The calculation gives \(RS = \frac{28}{5}\), which matches option (B).