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Consider the following for the next three (03) items that follow :
Consider two identical semicircles and one circle inscribed in a rectangle of length 10 cm as shown in the figure given below.
(Take $\pi = 3.14$ and $\sqrt{2} = 1.4$).

What is the area of triangle EOF?
 

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
12.5 square cm

To solve for the area of triangle EOF, let's first understand the context. We have a rectangle with two identical semicircles and a circle inscribed within it. Our goal is to find the area of the triangle EOF formed within this setup.

Assuming the given rectangle's length is 10 cm, the semicircles and inscribed circle will influence the dimensions.

  1. The diameter of each semicircle would be consistent with the width of the rectangle. Hence, the diameter of each semicircle is 10 cm. Therefore, the radius r is 5 \, \text{cm}.
  2. Now, let's assume the center of the circle and along with the ends of the diameter of one semicircle forms the triangle EOF. The circle's center is at the midpoint of the diameter of the semicircle because it is internally tangent.
  3. In triangle EOF, one side is the radius of the semicircle (5 cm), and the height from the endpoint of the radius to the other semicircle diameter is also the radius (5 cm).
  4. Since it's right-angled, we can solve the area using:

The area of triangle EOF is given by the formula for the area of a right-angle triangle:

\[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]
  • Here, both the base and height are equal to the radius of the semicircle, which is 5 cm.
  • Substituting the values, we get:

\[ \text{Area} = \frac{1}{2} \times 5 \, \text{cm} \times 5 \, \text{cm} = \frac{1}{2} \times 25 \, \text{cm}^2 = 12.5 \, \text{cm}^2 \]

Thus, the correct option is 12.5 square cm. The other options cannot be correct based on the geometric interpretation and calculations.

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