Question

in a triangle ABC AB = 9 BC is equal to 10 and AC is equal to 13 if G is centroid and D is the midpoint of BC then length of GD is?

Answer 1

Given:

In Δ ABC,

AB = 9

BC = 10

AC = 13

G is the centroid

D is the midpoint of BC

To find:

The length of GD

Solution:

Since D is the midpoint of BC

∴ BD = CD = 1/2 * BC = 1/2 * 10 = 5

Here in Δ ABC, AD is a median .... [∵ A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side of the triangle]

We know according to the Apollonius Theorem if AD is the median of ΔABC, then we have

$\boxed{\boxed{\bold{AB^2 + AC^2 = 2[AD^2 + BD^2]}}}$

By substituting the given values in the above theorem, we get

$AB^2 + AC^2 = 2[AD^2 + BD^2]$

$\implies 9^2 + 13^2 = 2[AD^2 + 5^2 ]$

$\implies 81 + 169 = 2[AD^2 + 25 ]$

$\implies250 = 2AD^2 + 50$

$\implies250 - 50 = 2AD^2$

$\implies200 = 2AD^2$

$\implies AD^2 = 100$

$\implies AD = \sqrt{100}$

$\implies AD = 10$

Now, we know that the centroid of a triangle divides each of its medians in the ratio of 2:1 i.e., $\frac{AG}{GD} = \frac{2}{1}$

∴ $\frac{GD}{AD} = \frac{GD}{AG + GD} = \frac{1}{1+2}$

$\implies \frac{GD}{AD} = \frac{1}{3}$

$\implies GD = \frac{1}{3} \times AD$

substituting AD = 10

$\implies GD = \frac{1}{3} \times 10$

$\implies\bold{ GD = \frac{10}{3}}$

Thus, the length of the GD is $\underline{\frac{10}{3}}$.

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