Question

In the figure, AP=12 cm, PB = 16 cm. Let \(\pi=3\), then the perimeter of the shaded portion is

• A. 52 cm
• B. 58 cm
• C. 56 cm
• D. 62 cm

Answer 1

The Correct Option is B

To solve the problem, we need to determine the perimeter of the shaded portion in the given figure. The shaded region consists of a semicircular arc and two straight line segments.

1. Understanding the Geometry:
The figure shows a semicircle with diameter \( AB \). The points \( A \) and \( B \) are endpoints of the diameter, and \( P \) is a point on the semicircle such that \( AP = 12 \, \text{cm} \) and \( PB = 16 \, \text{cm} \). The shaded region includes the semicircular arc from \( A \) to \( B \) and the two straight line segments \( AP \) and \( PB \).

2. Calculating the Diameter \( AB \):
Since \( AP \) and \( PB \) are segments of the semicircle, the total length of the diameter \( AB \) is: \[ AB = AP + PB = 12 + 16 = 28 \, \text{cm} \] Thus, the radius \( r \) of the semicircle is: \[ r = \frac{AB}{2} = \frac{28}{2} = 14 \, \text{cm} \]

3. Calculating the Length of the Semicircular Arc:
The circumference of a full circle is given by \( 2\pi r \). For a semicircle, the arc length is half of the circumference: \[ \text{Arc length} = \pi r \] Given that \(\pi = 3\), the arc length is: \[ \text{Arc length} = 3 \times 14 = 42 \, \text{cm} \]

4. Calculating the Perimeter of the Shaded Region:
The perimeter of the shaded region consists of the semicircular arc and the two straight line segments \( AP \) and \( PB \). Therefore, the total perimeter is: \[ \text{Perimeter} = \text{Arc length} + AP + PB \] Substituting the known values: \[ \text{Perimeter} = 42 + 12 + 16 = 70 - 12 = 58 \, \text{cm} \]

Final Answer:
The perimeter of the shaded portion is \({58 \, \text{cm}}\).