Question

Fill in the blanks.

Point G is the centroid of ∆ABC.

i. If l(RG) = 2.5, then l(GC) = ___ 

ii. If l(BG) = 6, then l(BQ) = ____ 

iii. If l(AP) = 6, then l(AG) = ___ and l(GP) = ___.

Answer 1

The centroid of a triangle divides each median in the ratio 2:1.

i. Point G is the centroid and seg CR is the median.

∴ l(GC) × 1 = 2 × 2.5 

∴ l(GC) = 5

ii. Point G is the centroid and seg BQ is the median.

∴ 3 = l(GQ) 

i.e. l(GQ) = 3

Now, l (BQ) = l(BG) + l(GQ) 

∴ l(BQ) = 6 + 3 

∴ l(BQ) = 9

iii. Point G is the centroid and seg AP is the median.

∴ l(AG) = 2 l(GP) …..(i)

Now, l(AP) = l(AG) + l(GP) … (ii) 

∴ l(AP) = 2l(GP) + l(GP) … [From (i)]

∴ l(AP) = 3l(GP)

∴ 6 = 3l(GP) ..[∵ l(AP) = 6]

\(\cfrac{6}{3}\) = l(GP)

∴ 2 = l(GP)

i.e. l(GP) = 2

l(AP) = l(AG) + l(GP) …[from (ii)]

∴ 6 = l(AG) + 2

∴ l(AG) = 6 – 2

∴ l(AG) = 4