Question

Area of the largest triangle that can be inscribed in a semi-circle of radius $r$ units is

• A. $r^2$ sq. units
• B. $\frac{1}{2}{r}^2$ sq. units
• C. $2r^2$ sq. units
• D. $\sqrt{2}{r}^2$ sq. units

Answer 1

The correct option is A $r^2$ sq. units

The area of a triangle is equal to the base times the height.
In a semi circle, the diameter is the base of the semi-circle.
This is equal to $2\times r$ (r = the radius)
If the triangle is an isosceles triangle with an angle of ${45}^{\circ }$ at each end, then the height of the triangle is also a radius of the circle.
A = $\frac{1}{2}\times b\times h$ formula for the area of a triangle becomes
A = $\frac{1}{2}\times 2\times r\times r$ because:
The base of the triangle is equal to $2\times r$
The height of the triangle is equal to r
A = $\frac{1}{2}\times 2\times r\times r$ becomes:
A = $r^2$