Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is
Let ABC be an isosceles triangle with AB = AC and a circle with cenre 0 and radius r, touching sides AB,BC,CA at D,F,E resp.
In let and
Now,
Squaring both sides, we get
Now, P (perimeter of )
Differentiate above equation w.r.t y, we get
for maxima and minima of p, put
Now, again differentiating w.r.t to y we get
Hence, perimeter p of is least for and least perimeter is
Hence proved