MATH SOLVE

4 months ago

Q:
# In the diagram, JG = 5 cm and GE = 10 cm. Based on this information, can G be a centroid of triangle HJK? Point G cannot be a centroid because JG does not equal GE. Point G cannot be a centroid because JG is shorter than GE. Point G can be a centroid because GE and JG are in the ratio 2:1. Point G can be a centroid because JG + GE = JE.

Accepted Solution

A:

Point G cannot be a centroid because JG is shorter than GE.
Without the diagram, this problem is rather difficult. But given what a centroid is for a triangle, let's see what statements make or do not make sense. Assumptions made for this problem.
G is a point within the interior of the triangle HJK.
E is a point somewhere on the perimeter of triangle HJK and that a line passing from that point to a vertex of triangle HJK will have point G somewhere on it.
Point G cannot be a centroid because JG does not equal GE.
* If G was a centroid, then JG would not be equal to GE because if that were the case, you could construct a circle that's both tangent to all sides of the triangle while simultaneously passing through a vertex of the triangle. That's impossible, so this can't be the correct choice.
Point G cannot be a centroid because JG is shorter than GE.
* This statement would be true. So this is a good possibility as the correct answer assuming the above assumptions are correct.
Point G can be a centroid because GE and JG are in the ratio 2:1.
* There's no fixed relationship between the lengths of the radius of a circle who's center is at the centroid and the distance from that center to a vertex of the triangle. And in fact, it's highly likely that such a ratio will not even be constant within the same triangle because it will only be constant of the triangle is an equilateral triangle. So this statement is nonsense and therefore a bad choice.
Point G can be a centroid because JG + GE = JE.
* Assuming that the assumption about point E above is correct, then this relationship would hold true for ANY point E on the side of the triangle that's opposite to vertex J. And only 1 of the infinite possible points is correct for the line JE to pass through the centroid. So this is also an incorrect choice.
Since of the 4 available choices, all but one are complete and total nonsense when speaking about a centroid in a triangle, that one has to be the correct answer. So "Point G cannot be a centroid because JG is shorter than GE."