5.1.2 Medians, Altitudes, and Midsegments in Triangles (continued)
The centroid of the triangle is the point of concurrency of the medians of a triangle. In other words, the a triangle’s centroid is the point at which the triangle’s three medians intersect. By the Centroid Theorem, the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side. This theorem can be used to find the length of a median. For example, suppose BP = 12. Then, since BP = (2/3) BZ , it follows that 12 = (2/3) BZ , and so BZ = 18.
Example 2 Finding the Equation of a Median
Remember, the equation of a line can be written using the coordinates of two points on that line. A median is a segment whose endpoints are a vertex of the triangle and the midpoint of the other side. In this example, the median passes through vertex C . So, the median passes through ( − 4, − 1). The side opposite of C is AB . So, the median must pass through the midpoint of AB . Use the midpoint formula to find the midpoint of AB : (0, 2). Now that two points on the median are known, ( − 4, − 1) and (0, 2), use the slope formula to find the slope of the line that passes through those two points: m = 3/4. Next, identify the y -intercept. Notice that the median passes through (0, 2), which is a point on the y -axis. Therefore, the y -intercept of the median is 2. Use the slope and the y -intercept to write the equation of the line.
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