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G is the centroid of the equilateral triangle ABC. If AB = 36 cm, then length (in cm) of AG is:
86
103
66
123
123
In an equilateral triangle, the distance from the centroid to a vertex is 32 of the height. Since the height is 23a, the distance $AG = \frac{2}{3} \times \frac{\sqrt{3}}{2}a = \frac{a}{\sqrt{3}}$$.
Side of the equilateral triangle AB = 36 cm, G is the centroid.
AG=3a, where a is the side length.
In an equilateral triangle, the distance from the centroid to a vertex is 32 of the height. Since the height is 23a, the distance $AG = \frac{2}{3} \times \frac{\sqrt{3}}{2}a = \frac{a}{\sqrt{3}}$$.
Students often mistake the distance from the centroid to the side (inradius) for the distance from the centroid to the vertex (circumradius).
Identify the formula for distance from vertex to centroid
In an equilateral triangle with side length a, the distance from the vertex to the centroid AG is given by the formula involving the height of the triangle.
AG=32×height=32×23a=3a
Substitute the given side length
Substitute a=36 cm into the derived formula.
AG=336
Rationalize and simplify
Multiply the numerator and denominator by 3 to rationalize the expression.
AG=336×3=123 cm
D is correct because the distance from the vertex to the centroid of an equilateral triangle with side 36 cm is 336=123 cm.
Remember that the centroid divides the median in a 2:1 ratio, which is a fundamental property applicable to any triangle.
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