△ABC

Question

△ABC
 is drawn on a coordinate plane with vertices A(1,3)
, B(6,6)
, and C(3,1)
 and with medians indicating the midpoint of each of the lines AB
, BC
, and CA
. Prove that the medians meet at a single point by finding the centroid. Express all results in fractions.(1 point)
centroid=
(
,
)
R

GPT-4o mini · Accepted Answer

To find the centroid of triangle $ ABC $ with vertices $ A(1,3) $, $ B(6,6) $, and $ C(3,1) $, we use the formula for the centroid $ G $ of a triangle given by the coordinates: $$ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) $$ where $ (x_1, y_1) $, $ (x_2, y_2) $, and $ (x_3, y_3) $ are the coordinates of the vertices $ A $, $ B $, and $ C $. Plugging in the coordinates: - $ A(1, 3) $: $ x_1 = 1, y_1 = 3 $ - $ B(6, 6) $: $ x_2 = 6, y_2 = 6 $ - $ C(3, 1) $: $ x_3 = 3, y_3 = 1 $ Now, we calculate the $ x $ and $ y $ coordinates of the centroid: $$ G_x = \frac{1 + 6 + 3}{3} = \frac{10}{3} $$ $$ G_y = \frac{3 + 6 + 1}{3} = \frac{10}{3} $$ Therefore, the coordinates of the centroid $ G $ are: $$ G\left(\frac{10}{3}, \frac{10}{3}\right) $$ Thus, the centroid is at: $$ \text{centroid} = \left(\frac{10}{3}, \frac{10}{3}\right) $$ This point confirms that the medians of the triangle intersect at a single point (the centroid).