Solve the problem.
-The medians of a triangle intersect at a point. The distance from the vertex to the point is exactly two-thirds of the distance from the vertex to the midpoint of the opposite side. Find the exact distance of that point from the
Vertex A(3, 4) of a triangle, given that the other two vertices are at (0, 0) and (8, 0).
A) $\frac { 8 } { 3 }$
B) $\frac { \sqrt { 17 } } { 3 }$
C) $\frac { 2 \sqrt { 17 } } { 3 }$
D) 2

Question

Quizplus · Accepted Answer

The centroid (the point where the medians intersect) divides each median into two segments, one of which is twice the length of the other. The longer segment, from the vertex to the centroid, is two-thirds the length of the entire median. The distance from A(3, 4) to the midpoint of the opposite side (which is (4, 0) since the other vertices are at (0, 0) and (8, 0)) can be found using the distance formula: $\sqrt{(3-4)^2 + (4-0)^2} = \sqrt{1^2 + 4^2} = \sqrt{17}$. Therefore, two-thirds of this distance is $\frac{2}{3} \sqrt{17}$, which matches choice C.

Solve the Problem 83\frac { 8 } { 3 }38​ B) 173\frac { \sqrt { 17 } } { 3 }317​​

Solve the Problem $\frac { 8 } { 3 }$ B) $\frac { \sqrt { 17 } } { 3 }$