Copy of 3.8 The Centroid (Investigation 1)


3.8 The Centroid (Investigation 1)

Step 1: On a sheet of patty paper, draw as large a scalene triangle as possible and label it CNR, as shown at right. Locate the midpoints of the three sides. Construct the medians and complete the conjecture. Median Concurrency Conjecture The three medians of a triangle are concurrent Step 2: Label the three medians line segment CD, NO, and RE. Label the centroid D. Step 3: Use your compass or another sheet of patty paper to investigate whether thee is anything special about the centroid. Is the centroid equidistant from the three vertices? From the three sides? Is the centroid the midpoint of each median? The centroid is not equidistant from the three vertices however it is from the three sides to the centroid Step 4: The centroid divides a median into two segment . Focus on one median. Use you patty paper or compass to compare the length of the longer segment to the length of the shorter segment and find the ratio. One median I'm focusing on is FC where the ratio would be CG/FG or in numerical form, 5.9/2.95  where it would then be 2/1  simplified. Step 5: Find the ratios of the lengths of the segment parts for the other two medians. Do you get the same ratio for each median? The ratios of the lengths of the segment parts for the other two medians would be 3.24/1.62 and 4.92/2.46 where simplified would be 2/1  and yes they do have the same ratio for each median. Centroid Conjecture The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side.