A triangle has corners at #(3 ,7 )#, #(2 ,5 )#, and #(8 ,4 )#. What is the area of the triangle's circumscribed circle?
2 Answers
Area of the triangle's circumscribed circle:
Explanation:
Perhaps there is a simpler way, but here is how I would do it:
Part 1: Find the Lengths of the Three Sides
Let
Let
Let
Part 2: Find the Perimeter, Semi-Perimeter, and Area of the Triangle
(I will need the help of a calculator here)
Perimeter:
Semi-perimeter:
Area of triangle (using Heron's formula):
Part 3: Find the Radius and Area of the Circumscribed Circle
(more calculator usage)
Radius of Circumscribed Circle:
(ask as a separate question if you are unfamiliar with this relation).
Area of Circle:
Explanation:
The standard form for the equation of a circle is:
where
Because the triangles vertices are points on the circumscribed circle, we can use the 3 points and the standard form to write 3 equations:
We have 3 equations and we can use them to find the values of, h, k, and
Temporarily eliminate
Expand the squares, using the pattern,
Please notice that for every
Combine all of the constant terms into a single term on the right:
Combine all of the h terms into a single term on the right:
Combine all of the k terms into a single term on the left:
Multiply equation [4] by -3 and equation [5] by 2 and add them:
Substitute
Substitute the values for h and k into equation [3]
The area of the circle is