For Exercise, you’ll need to recall the following definitions and results from elementary geometry. In a triangle, a line segment drawn from a vertex to the midpoint of the opposite side is called a median. The three medians of a triangle are concurrent; that is, they intersect in a single point. This point of intersection is called the centroid of the triangle. A line segment drawn from a vertex perpendicular to the opposite side is an altitude. The three altitudes of a triangle are concurrent; the point where the altitudes intersect is the orthocenter of the triangle.
This exercise illustrates the fact that the altitudes of a triangle are concurrent. Again, we’ll be using ∆ABC with vertices A(−4, 0), B(2, 0), and C(0, 6). Note that one of the altitudes of this triangle is just the portion of the y-axis extending from y = 0 to y = 6; thus, you won’t need to graph this altitude; it will already be in the picture.
(a) Using paper and pencil, find the equations for the three altitudes. (Actually, you are finding equations for the lines that coincide with the altitude segments.)
(b) Use a graphing utility to draw ∆ABC along with the three altitude lines that you determined in part (a). Note that the altitudes appear to intersect in a single point. Use the graphing utility to estimate the coordinates of this point.
(c) Using simultaneous equations (from intermediate algebra), find the exact coordinates of the orthocenter. Are your estimates in part (b) close to these values?
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