If a right angled △ A BC of maximum area is inscribed within a circle of radius R , then ( Δ represents area of triangle ABC and r , r 1 , r 2 , r 3 represent inradius and exradii, and s is the

Question

If a right angled △ A BC of maximum area is inscribed within a circle of radius R , then ( Δ represents area of triangle ABC and r , r 1 ​ , r 2 ​ , r 3 ​ represent inradius and exradii, and s is the semi perimeter of △ A BC then

TestHub · Accepted Answer

For a right-angled triangle inscribed in a circle of radius $R$ , the length of the hypotenuse is $2 R$ . The area is maximum when it is an isosceles triangle, with each side equal to $2 R$ .

$S Δ r \frac{1}{r _{1}} + \frac{1}{r _{2}} + \frac{1}{r _{3}} = \frac{1}{2} (22 R + 2 R) = (2 + 1) R = \frac{1}{2} (2 R) (2 R) = R^{2} = \frac{Δ}{S} = \frac{R ^{2}}{( 2 + 1 ) R} \Rightarrow r = (2 - 1) R = \frac{1}{r} = \frac{1}{( 2 - 1 ) R} = \frac{2 + 1}{R}$

Mathematics - Properties of Triangle Question with Solution | TestHub

Options:(select one or more)

Doubts & Discussion