GATE DA 2026 Set 1 — Question 10
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
In the given figure, , , and are three points on a circle of radius 10 cm with as its center, , and . The figure is representative.
The area of the shaded region is ______________ cm.
The area of the shaded region is ______________ cm.
Correct answer
(C) 50√(2)
Solution
Given:
Given , we have:
In , (since and are on the circle).
Thus, is an isosceles triangle.
The angle at the center, , is:
Similarly, for , .
Here, sides are radii .
Since , . Total Area of
Alternatively, using kite properties:
- Radius cm.
- is the center.
- are on the circle.
- .
- .
1.Analyze the triangles:
Consider and .- (radii).
- (common side).
- (given).
2.Find angles:
Since the triangles are congruent, .Given , we have:
In , (since and are on the circle).
Thus, is an isosceles triangle.
The angle at the center, , is:
Similarly, for , .
3.Calculate Area:
The area of a triangle with two sides and included angle is .Here, sides are radii .
Since , . Total Area of
Alternatively, using kite properties:
- Diagonals are and .
- (radius).
- Central angle . So is a right isosceles triangle.
- Chord .
- Area of kite with perpendicular diagonals = .
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