A triangle has corners at #(3 ,5 )#, #(4 ,9 )#, and #(7 ,6 )#. What is the area of the triangle's circumscribed circle?
1 Answer
The area of the circumscribed circle is:
Explanation:
When I do this type problem, I always shift all of the corners so that one of them is the origin. This does not change the area of the circumscribed circle and it makes one of the 3 equations (that we must write) become very simple and useful.
Subtract 3 from every x coordinate and subtract 5 from every y coordinate:
Use the standard form of the equation of a circle,
, and the 3 shifted points to write three equations:
Please observe that equation [1] simplifies to:
This allows us to substitute
Expand the squares, using the pattern #(a - b)^2 = a^2 - 2ab + b^2:
collect the constant terms into a single term on the right:
Multiply equation [8] by -4 and add to equation [7]:
Substitute 1.7 for h in equation [8]:
Use equation [4] to find the value of
The area of a circle is:
The area of the circumscribed circle is: