The question asks for the area of a circular segment. A segment is the region between a chord and the arc it cuts off.

Calculating Circle Segment Area

To find the area of the segment, we subtract the area of the triangle formed by the radii and the chord from the area of the sector formed by the same radii and the arc.

Area of Segment = Area of Sector - Area of Triangle

Area of the Sector

The formula for the area of a sector is:

$ \text{Area}_{\text{Sector}} = \frac{\theta}{360^\circ} \times \pi r^2 $

Given:

• Radius ($r$) = $10\text{ cm}$
• Central angle ($\theta$) = $90^\circ$

Substituting the values:

$ \text{Area}_{\text{Sector}} = \frac{90^\circ}{360^\circ} \times \pi (10)^2 $

$ \text{Area}_{\text{Sector}} = \frac{1}{4} \times 100\pi $

$ \text{Area}_{\text{Sector}} = 25\pi \text{ cm}^2 $

Area of the Triangle

The triangle formed by the two radii and the chord is an isosceles triangle with the angle between the equal sides (radii) being $90^\circ$. This means it's a right-angled isosceles triangle.

The area of this triangle can be calculated using the formula:

$ \text{Area}_{\text{Triangle}} = \frac{1}{2} \times \text{base} \times \text{height} $

In this case, the two radii act as the base and height.

$ \text{Area}_{\text{Triangle}} = \frac{1}{2} \times r \times r $

$ \text{Area}_{\text{Triangle}} = \frac{1}{2} \times 10 \times 10 $

$ \text{Area}_{\text{Triangle}} = \frac{1}{2} \times 100 $

$ \text{Area}_{\text{Triangle}} = 50 \text{ cm}^2 $

Area of the Segment

Now, subtract the triangle's area from the sector's area:

$ \text{Area}_{\text{Segment}} = \text{Area}_{\text{Sector}} - \text{Area}_{\text{Triangle}} $

$ \text{Area}_{\text{Segment}} = 25\pi - 50 \text{ cm}^2 $

This result matches option A.