ABC has the vertices A(1, 4), B(3, 4), and C(1, 1). Find the coordinates of each point of concurrency. 13. circumcenter of ∆ABC 14. orthocenter of ∆ABC
13. Circumcenter is where the perpendicular bisectors of the sides of the triangles meet. Here AB is parallel to the x-axis and AC is parallel to the y-axis and so it is easy to find the perpendicular bisectors of these two sides. The midpoint of AB = ( (1+3)/2, (4+4)/2) ) = (2, 4). The equation of the perpendicular bisector of AB is x = 2 The midpoint of AC = ( (1+1)/2, (1+4)/2) ) = (1, 2.5). The equation of the perpendicular bisector of AC is y = 2.5 These two perpendicular bisectors will meet at (2, 2.5) which is the circumcenter. 14. Orthocenter is where the altitudes meet. In a right triangle, the altitude of side AB is AC and the altitude of side AC is AB. And they meet at A. So for a right triangle the orthocenter is at the vertex that has the 90 degrees. Orthocenter = (1,4).