AMSCO'S Geometry. New York - Rye High School

Concurrence of the Altitudes of a Triangle 319The point where the altitudes of a triangle intersect is called the orthocenter.EXAMPLE 1The coordinates of the vertices of PQR are P(0, 0), Q(2, 6), and R(4, 0). Findthe coordinates of the orthocenter of the triangle.Solution Let PL be the altitude from P to QR.The slope of QR 6 2 0is 22 2 4 5 26 6 521 .The slope of PL is 1.y 2 0The equation of PL is x 2 0 5 1 ory x.Let QN be the altitude from Q to PR.The point of intersection, N, is on theline determined by P and R.The slope of PR is 0 since PR is a horizontalline. Therefore, QN is a segmentof a vertical line that has no slope.The equation of QN is x 2.Q(2, 6)The intersection S of the altitudes QN and PL is the common solution of theequations x 2 and y x. Therefore, the coordinates of the intersection Sare (2, 2). By Theorem 8.4, point S is the orthocenter of the triangle or thepoint where the altitudes are concurrent.NS(2, 2)yLP(0, 0)MR(4, 0)AnswerAlternativeSolutionThe orthocenter of PQR is S(2, 2).Use the result of the proof given in this section.The coordinates of the point of intersection ofthe altitudes are A 0, 2 acb B . In order for this resultto apply, Q must lie on the y-axis and P and Rmust lie on the x-axis. Since P and R already lieon the x-axis, we need only to use a translation tomove Q in the horizontal direction to the y-axis.Use the translation (x, y) → (x + 2, y):Therefore,P(0, 0) → P(2, 0) Q(2, 6) → Q(0, 6) R(4, 0) → R(6, 0)A(a, 0) P(2, 0) or a 2B(0, b) Q(0, 6) or b 6C(c, 0) R(6, 0) or c 6.yQ(2, 6) Q(0, 6)R(6, 0)P(0, 0) P(2, 0) R(4, 0) x