The Perimeter Of The Shaded Region In The Given Figure Is (A) L (B) L+ A (C) L+ 2r (D) L+2r+a

Mar 10, 2025 by ADMIN 96 views

18. The Perimeter of the Shaded Region in the Given Figure

Introduction

The perimeter of a shape is the total length of its boundary. In the given figure, we are asked to find the perimeter of the shaded region. To solve this problem, we need to carefully analyze the figure and identify the different components that make up the perimeter of the shaded region.

Understanding the Figure

The given figure consists of a rectangle with a semicircle attached to one of its sides. The shaded region is the area enclosed by the semicircle and the rectangle. To find the perimeter of the shaded region, we need to consider the lengths of the different segments that make up its boundary.

Identifying the Components of the Perimeter

The perimeter of the shaded region consists of three main components:

The length of the rectangle (l)
The radius of the semicircle (r)
The length of the arc of the semicircle (a)

Calculating the Perimeter

To calculate the perimeter of the shaded region, we need to add up the lengths of these three components. The length of the rectangle is given as l, the radius of the semicircle is given as r, and the length of the arc of the semicircle is given as a.

Analyzing the Options

Now, let's analyze the options given in the problem:

Option (A): l
Option (B): l + a
Option (C): l + 2r
Option (D): l + 2r + a

Conclusion

To determine the correct answer, we need to carefully consider the lengths of the different components that make up the perimeter of the shaded region. The length of the rectangle is l, the radius of the semicircle is r, and the length of the arc of the semicircle is a.

Final Answer

The correct answer is option (D): l + 2r + a. This is because the perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (which is twice the radius, 2r), and the length of the arc of the semicircle (a).

Explanation

The diameter of the semicircle is twice the radius, so it is equal to 2r. The length of the arc of the semicircle is equal to the circumference of the semicircle, which is half the circumference of the full circle. The circumference of the full circle is 2πr, so the circumference of the semicircle is πr. However, the length of the arc of the semicircle is not equal to πr, but rather a portion of it. To find the length of the arc of the semicircle, we need to consider the angle subtended by the arc at the center of the circle. However, in this case, we are not given any information about the angle subtended by the arc, so we cannot determine the length of the arc of the semicircle.

Alternative Solution

However, we can still find the perimeter of the shaded region by considering the lengths of the different segments that make up its boundary. The perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a). However, we are not given any information about the length of the arc of the semicircle, so we cannot determine the perimeter of the shaded region.

Conclusion

In conclusion, the correct answer is option (D): l + 2r + a. This is because the perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a). However, we are not given any information about the length of the arc of the semicircle, so we cannot determine the perimeter of the shaded region.

Final Answer

The final answer is option (D): l + 2r + a.

Introduction

In the previous article, we discussed the problem of finding the perimeter of the shaded region in the given figure. We analyzed the different components that make up the perimeter and concluded that the correct answer is option (D): l + 2r + a. However, we also noted that we are not given any information about the length of the arc of the semicircle, which makes it difficult to determine the perimeter of the shaded region.

Q&A

Here are some frequently asked questions about the problem:

Q: What is the perimeter of the shaded region?

A: The perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a).

Q: How do we find the length of the arc of the semicircle?

A: We are not given any information about the angle subtended by the arc at the center of the circle, so we cannot determine the length of the arc of the semicircle.

Q: What is the correct answer?

A: The correct answer is option (D): l + 2r + a.

Q: Why is the correct answer option (D)?

A: The perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a). Therefore, the correct answer is option (D): l + 2r + a.

Q: What is the diameter of the semicircle?

A: The diameter of the semicircle is twice the radius, so it is equal to 2r.

Q: What is the circumference of the semicircle?

A: The circumference of the semicircle is half the circumference of the full circle, which is πr.

Q: What is the length of the arc of the semicircle?

A: We are not given any information about the angle subtended by the arc at the center of the circle, so we cannot determine the length of the arc of the semicircle.

Conclusion

In conclusion, the perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a). However, we are not given any information about the length of the arc of the semicircle, so we cannot determine the perimeter of the shaded region. The correct answer is option (D): l + 2r + a.

Final Answer

The final answer is option (D): l + 2r + a.

Additional Tips

To find the perimeter of the shaded region, you need to consider the lengths of the different segments that make up its boundary.
The perimeter of the shaded region consists of the length of the rectangle (l), the diameter of the semicircle (2r), and the length of the arc of the semicircle (a).
However, we are not given any information about the length of the arc of the semicircle, so we cannot determine the perimeter of the shaded region.

Common Mistakes

Not considering the lengths of the different segments that make up the perimeter of the shaded region.
Not understanding the concept of the diameter of the semicircle.
Not understanding the concept of the circumference of the semicircle.