3. Point Ris the centroid of Delta ABC,AR=8,RC=12,BD=10 and BE=15. Find each indicated measure. The point of con concurrency of the 3 medians is called the square A median is a segment that conny vertex to the square of the opposite side. possible answers for statement above: (centroid circumcenter, incenter, orthocenter , midpoint, side, vertex, ongle) DR= BC= BR=

<p> The question provides that point R is the centroid of triangle ABC which divides the medians into the ratios of 2:1. It also mentions some measure for two more line segments BD and BE. It finally asks for specific measures in triangle like segment DR, BC and BR.<br /><br />a) In a triangle, the properties of the centroid is such that it divides the median (he line from a vertex to the midpoint of the opposite side) in a ratio of 2:1. Here since \(AR = 8 \) and \( RC = 12 \), these already are in the ratio 2:1 implying 'R' is two thirds the distance from vertex 'A' to the midpoint of BC.<br /> <br />b) The point of concurrency (point where they intersect) for medians in a triangle is called centroid. In here, for completion of the given statement in the problem, we will select the empty squares to be "centroid" and "midpoint". A median is a segment that connects a vertex to the "midpoint" of the opposite side.<br /> <br />c) \( DR = \square \) - Since BD is given as 10 units and DR is one third of BD (property of centroid), so, \( DR = \frac{1}{3} \times BD = \frac{1}{3} \times 10 = 10/3\) <br /> <br />d) \(BC = \square \) - Since BE is marked as 15units, which is two thirds of the length (lines are in ratio 2:1), hence \( BC = 1.5 \times BE = 1.5 \times 15 = 22.5 \)<br /><br />e) \( BR = \square \) - Now BR = BD - DR = 10 - 10/3 = 20/3<br /> <br /> Using the principles and properties of centroids, and magnitudes, we have these values. </p>