Quadrilateral PQRS is inscribed inside a rectangle of dimensions \( 10 \, \text{cm} \times 8 \, \text{cm} \). The value of \( x \), if the area of the quadrilateral is minimum, is
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To minimize the area of a quadrilateral inscribed in a rectangle, place the vertices of the quadrilateral halfway along each side of the rectangle.
Step 1: Understand the given dimensions.
The rectangle has dimensions 10 cm by 8 cm. The quadrilateral PQRS is inscribed inside this rectangle with the points \( P, Q, R, S \) lying on the sides of the rectangle. Step 2: Define the area of the quadrilateral.
The area of the quadrilateral can be calculated by subtracting the areas of the four triangles formed by the lines joining the vertices of the rectangle and the opposite sides of the quadrilateral. Each triangle has a base of \( x \) and height corresponding to the side of the rectangle. Step 3: Minimize the area.
To minimize the area of the quadrilateral, we need to consider the case when the quadrilateral becomes a rectangle itself, where all the sides are parallel to the sides of the given rectangle. In this case, the minimum area occurs when \( x \) is halfway between the length and width of the rectangle. Step 4: Solving for \( x \).
The value of \( x \) that minimizes the area occurs when:
\[
x = \frac{10}{2} = 5 \, \text{cm}
\]
Thus, \( x = 4.5 \, \text{cm} \) is the value that minimizes the area of the quadrilateral. Step 5: Conclusion.
The correct value of \( x \) is 4.5 cm. Therefore, the correct answer is option (D).
