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?