SOLUTION: Find the coordinates of the centroid, orthocenter, and circumcenter of a triangle with vertices A(4, -1), B(2, 6), and C(9, -5).

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Question 524163: Find the coordinates of the centroid, orthocenter, and circumcenter of a triangle with vertices A(4, -1), B(2, 6), and C(9, -5).

Found 2 solutions by solver91311, richard1234:
Answer by solver91311(24713) (Show Source):
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Centroid:

Calculate the coordinates of the mid-points of AC and BC.

Use the mid-point formulas:

and

Write an equation for a line that passes through point A and the mid-point of BC.

Use the two-point form of an equation of a line:

where and are the coordinates of the given points.

Then write an equation for a line that passes through point B and the mid-point of AC

Using any convenient method, solve this 2X2 system. The solution set will be the centroid of the triangle -- the point of intersection of the three triangle medians.

Orthocenter:

Calculate the slope of sides AB and BC of the triangle using the slope formula:

where and are the coordinates of the given points.

Then, using the point-slope form of an equation, and the fact that perpendicular lines have slopes that are negative reciprocals, write equations of the two altitudes to sides AB and BC -- lines perpendicular to AC and passing through B and perpendicular to BC and passing through A.

Solve the 2X2 system. The intersection of the two altitudes is the orthocenter.

Circumcenter

Using the slopes calculated above for AB and BC and the mid-points calculated for the Centroid solution, write equations of the perpendicular bisectors of AB and BC. Perpendicular to AB and passing through the mid-point of AB, then perpendicular to BC and passing through the mid-point of BC.

Solve the 2X2 system. The intersection of the perpendicular bisectors is the circumcenter (a point equidistant from the three vertices and therefore the center of a circle that passes through the three vertices of the triangle.)

John

My calculator said it, I believe it, that settles it

Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website!
The easiest way to find the centroid is to take the average of the x-coordinates and the average of the y-coordinates (this corresponds to a "center of mass" encountered in physics). To find the orthocenter and circumcenter, use the methods that the other tutor suggested.