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G is the centroid of the equilateral triangle PQR. If PQ = 18 cm, then length (in cm) of PG is:
43
63
83
93
63
In an equilateral triangle, the distance from the centroid to a vertex is always 31 times the side length, or effectively the side length divided by 3.
Equilateral triangle PQR with side length PQ = 18 cm, where G is the centroid.
PG=3a
In an equilateral triangle, the distance from the centroid to a vertex is always 31 times the side length, or effectively the side length divided by 3.
Students often mistake the distance from the centroid to the side (inradius) with the distance to the vertex (circumradius). The distance to the side is 23a, while to the vertex is 3a.
Identify triangle properties
For an equilateral triangle with side length a, the altitude h is given by h=23×a.
h=23×18=93
Use Centroid Property
The centroid G divides the median (which is also the altitude in an equilateral triangle) in the ratio 2:1. The length of the segment PG from vertex to centroid is 32 of the altitude.
PG=32×h
Calculate PG
Substitute the value of h into the ratio formula to find PG.
PG=32×93=63
B is correct because the distance from the centroid to the vertex is calculated as 32×(23×18), resulting in 63 cm.
This concept of the 2:1 centroid ratio is frequently used in coordinate geometry to find the coordinates of the centroid given the vertices of a triangle.
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