SOLUTION: Analytic Geometry 8. c) Given only the three vertices of a triangle. Explain how you would find the coordinate of teh circumcentre of a triangle. Include a diagram to illustrate y

Question 189358: Analytic Geometry
8. c) Given only the three vertices of a triangle. Explain how you would find the coordinate of teh circumcentre of a triangle. Include a diagram to illustrate your answer.
Thank you very much!!!!
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

The circumcenter of a triangle is the point where the three perpendicular bisectors of the three sides intersect. That point is equidistant from each of the vertices so if a circle is constructed with center at the circumcenter and radius equal to the distance from the circumcenter to any of the vertices of the triangle, the circle will intersect the other two vertices as well.

Step 1: Using any two of the vertex points and the two-point form of the equation of a line, derive the equation for the line containing the segment defining the side of the triangle that has the two selected vertices as end points.

Step 2: Using a different pair of vertex points, repeat step 1.

Step 3: Using the mid-point formulas, determine the coordinates of the mid-point of the first line segment.

and

Step 4: Use:

to calculate the slope of a perpendicular to that side of the triangle.

Step 5: Use the point-slope form of the equation of a line to derive the equation of the perpendicular bisector of that side of the triangle.

Step 6: Repeat steps 3, 4, and 5 beginning with the equation derived in step 2.

Step 7: Solve the system of equations consisting of the equations resulting from steps 5 and 6. The solution set will consist of the ordered pair defining the circumcenter.

John

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