The diagram shows FGH, its medians, centroid P, and the lengths of some of the subsegments. Apply the Centroid Theorem to find other lengths.

The other lengths using the centroid theorem gives us;

FH = 24

PF = 8

GX = 13.5

         The centroid of a triangle is defined as the point where the three medians intersect each other.

Now, the centroid theorem states that the centroid is ²/₃ of the distance from each vertex to the midpoint of the opposite side.

Thus, applying the centroid theorem to the given triangle gives;

FH = 2(HX)

We see that HX = 12.

Thus; FH = 2 * 12

FH = 24

Also; PF = 2(PZ)

We see that; PZ = 4

Thus;

PF = 2(4)

PF = 8

Also, GP = 2(PX)

We have GP = 9.

Thus;

9 = 2(PX)

PX = 9/2

PX = 4.5

GX = GP + PX

GX = 9 + 4.5

GX = 13.5

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Step-by-step explanation:

We will use two properties to solve this problem,

1). Medians of a triangle bisect the sides of a triangle.

2). Centroid of a triangle divides the medians in the ratio of 2 : 1.

Length of FH = 2 × Length of HX

                      = 2 × 12

                FH = 24

PF : PZ = 2 : 1

PF = 8

GX = GP + PX

GP = 9  [Given]

Since,

PX = = 4.5

Therefore, GX = 9 + 4.5

GX = 13.5