Question

the area of largest triangle that can be inscribed in a semicircle of radius R centimetre is
A) r^2
B)(r/2)^2
C)r root 2
D)r​

Answer 1

option b is correct

Answer 2

Let the radius of the semicircle be 'r'

We know that angle subtended by a diameter to any point of a circle is 90°. So angle BAC=90°.

Using Pythagoras in ∆ABC,

$AB = \sqrt{ {r}^{2} + {r}^{2} } = \sqrt{2 {r}^{2} } = r \sqrt{2}$

Similarly, AC= $r\sqrt{2}$

Now,

$Area \: of \: the \: triangle = \frac{1}{2} bh = \frac{1}{2} \times r \sqrt{2} \times r \sqrt{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{2 {r}^{2} }{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {r}^{2}$

So the correct answer is option (a) ${r}^{2}$