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Fill in the blanks.

Point G is the centroid of ABC.

(1) If l(RG) = 2.5 then l(GC) = _____

(2) If l(BG) = 6 then l(BQ) = _____

(3) If l(AP) = 6 then l(AG) = _____ and l(GP) = _____

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#### Solution

In ∆ABC, the medians AP, BQ and CR to the sides BC, CA and AB respectively intersect at G. Since, centroid of a triangle divides the medians in the ratio of 2 : 1, then AG : GP = BG : GQ = CG : GR = 2 : 1.

(1) We have, CG : GR = 2 : 1

⇒ `(GC)/(RG) = 2/1`

⇒ `(GC)/(2.5) = 2/1`

⇒ GC = 5

(2) We have, BG : GQ = 2 : 1

⇒ `(BG)/(GQ) = 2/1`

⇒ `(BG)/(BQ-BG) = 2/1`

⇒ `(6)/(BQ-6) = 2`

⇒ BQ - 6 = 3

⇒ BQ = 9

(3) We have, AG : GP = 2 : 1

⇒ `(AG)/(GP) = 2/1`

⇒ `(AG)/(AP-AG) = 2`

⇒ `(AG)/(6-AG)` = 2

⇒ 2(6 - AG) = AG

⇒ 12 - 2AG = AG

⇒ 3AG = 12

⇒ `AG = 12/3`

⇒ AG = 4

Now, GP = AP − AG = 6 − 4 = 2.

Hence, we have,

(1) If l(RG) = 2.5 then l(GC) = 5

(2) If l(BG) = 6 then l(BQ) = 9

(3) If l(AP) =6 then l(AG) = 4 and l(GP) = 2

#### Notes

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