Question

how to find maximum area of triangle inscribed in a semicircle

Answer 1

Let the Radius is the semicircle be r ;

Maximum area of a triangle inscribed in a semicircle will be a isosceles right angled triangle with :

Dimeter of circle = Hypotenuse
H = 2r
other two sides = a

a^2 + a^2 = (2r)^2
2a^2 = 4r^2
a = √2 r
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Area of triangle

= (1/2) × b × h
= (1/2) × √2 r × √2 r
= (1/2) × 2 × r^2
= r^2
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Answer 2

The maximum area of triangle inscribed in a semicircle:
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Lets think the radius is r

The area can be maximum when,

The base= the diameter

or,B= 2r [ as the base sits on the diameter of the semicircle]

And the height = the radius

or,H=r

Now,

Area of the triangle is = 1/2 * B * H

                                       = 1/2 * 2r *r

                                       = r^2

Hope it helps.