Question

Suppose $\triangle ABC$ is an isosceles triangle with $\angle C = 90^\circ$, $A = (2, 3)$ and $B = (4, 5)$. Then the centroid of the triangle is

• A. $\left(\dfrac{13}{8}, \dfrac{8}{3}\right)$
• B. $\left(\dfrac{11}{3}, \dfrac{10}{3}\right)$
• C. $\left(\dfrac{10}{3}, \dfrac{13}{3}\right)$
• D. $\left(\dfrac{10}{3}, \dfrac{11}{3}\right)$

Hint:

Use right-angle and isosceles triangle properties to determine coordinates and then compute centroid.

Answer 1

The Correct Option is D

Given $\angle C = 90^\circ$ and triangle is isosceles, so $C$ is at the right angle vertex
Then $AC \perp BC$ and $AC = BC$
Let $C = (x, y)$
Use slopes: $(y - 3)/(x - 2) \cdot (y - 5)/(x - 4) = -1$ for perpendicularity
Also use $AC^2 = BC^2$ to find $C$
Then use centroid formula:
$G = \left( \dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3} \right) = \left( \dfrac{2 + 4 + 15}{3}, \dfrac{3 + 5 + 25}{3} \right) = \left( \dfrac{10}{3}, \dfrac{11}{3} \right)$