Final answer:
To determine the distances between each side of triangle ABC and the centroid, calculate the centroid's coordinates by averaging the x and y coordinates of the triangle's vertices. Then use the distance formula or perpendicular distances to find the required measurements.
Step-by-step explanation:
To find the point of intersection of the medians of triangle ABC with vertices A=(6,−3), B=(−4,5), and C=(−1,−2), we first need to calculate the midpoint of each side to construct the medians. A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side. The point where all three medians intersect is known as the centroid and can be found by averaging the x and y coordinates of the vertices of the triangle.
The midpoint of side AB, for instance, is obtained by averaging the x-coordinates and the y-coordinates of points A and B separately. The centroid (G) will have coordinates G=(x_G, y_G), where x_G=(x_A + x_B + x_C)/3 and y_G=(y_A + y_B + y_C)/3.
Once we find the centroid, we can calculate the distances from the centroid to each side of the triangle using the distance formula or by dropping perpendiculars from the centroid to the equations of the lines containing the sides of the triangle. This would give us the required distance between each side of the triangle and the point where the medians intersect.