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Question
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is ______.
Options
r2 sq.units
r2
2r2
`1/2 "r"^2` sq.units
2r2 sq.units
r3
`sqrt(2)"r"^2` sq.units
2r3
Solution
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is r2 sq.units.
Explanation:
The largest triangle that can be inscribed in a semi-circle of radius r units is the triangle having its base as the diameter of the semi-circle and the two other sides are taken by considering a point C on the circumference of the semi-circle and joining it by the endpoints of diameter A and B.
∴ ∠C = 90° ...(By the properties of circle)
So, ΔABC is right-angled triangle with base as diameter AB of the circle and height be CD.
Height of the triangle = r
∴ Area of largest ΔABC = `1/2 xx "Base" xx "Height"`
= `1/2 xx "AB" xx "CD"`
= `1/2 xx 2"r" xx "r"`
= r2 sq.units
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