A circle of radius
, is inscribed within an isosceles triangle
.
. Given that
is acute, find the ratio of the area of the circle to that of triangle
.
Mathematics Specialist 3AB Question15 page 55
I came across this question while searching for an area of sectors and segments question.
Here’s a diagram

We know are tangents to the circle. Because the triangle is isosceles, the distance from
to the circle is the same as the distance from
to the circle.

is perpendicular to
(because the triangle is isosceles).
Because it is proportional, i.e. and
, we can let
Let
(1)
but also equals
(2)
Set equation equal to equation
Square both sides
Expand and simplify
Divide by
Square both sides
I solved this using a graphics calculator
We can reject ,
, and
. If
, there isn’t a triangle, and if
is not acute.
Hence the area of triangle
(3)
(4)
Hence, equation divided by equation
is
Perhaps I approached this question in the wrong way. Is there an easier process?