# A triangle has corners at #(5 ,6 )#, #(4 ,3 )#, and #(2 ,2 )#. What is the area of the triangle's circumscribed circle?

##### 1 Answer

To solve use the following:

- Area of circle

- Equation of circle

- Know that all corners are points of the circle, so they satisfy the

equation

Answer is:

#### Explanation:

The area of the circle is:

Where

Where **including the three corners of the triangle** . Therefore, we have three equations:

- For point
#(5,6)#

By expanding the identities:

**Equation (1)**

- For point
#(4,3)#

By expanding the identities:

**Equation (2)**

- For point
#(2,2)#

By expanding the identities:

**Equation (3)**

Now from these three equations we can find

- Equations (1) minus (2) results in:

**Equation (4)**

- Equations (2) minus (3) results in:

**Equation (5)**

Substitute equation (4) in (5):

**Equation (6)**

Substitute equation (6) in equation (4) to find

Now that the coordinates of the circles center are known, the equation of the circle can be taken for any of the three known points. Let's pick for example point (2,2):

Finally, the area of the circle is: