Multiple choice

In $\triangle{ABC}$, $AB=9$cm , $BC=10$cm and $AC=13$cm.If $G$ is centroid of $\triangle{ABC}$ and $D$ is midpoint of $BC$ then the length of $GD$ is

  1. $25$
  2. $20$
  3. $\dfrac{10}{3}$
  4. $\dfrac{20}{3}$
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C Correct answer
Explanation

In triangle ABC, D is the midpoint of BC, so AD is the median to side BC. Using Apollonius theorem: AB^2 + AC^2 = 2(AD^2 + BD^2). With AB=9, AC=13, BC=10 (BD=5): 81 + 169 = 2(AD^2 + 25) => 250 = 2(AD^2 + 25) => 125 = AD^2 + 25 => AD^2 = 100 => AD = 10. The centroid G divides the median AD in ratio 2:1, so GD = 1/3 * AD = 10/3.

AI explanation

To find the length of GD, we first find the median AD using Apollonius's theorem, which states that AB squared plus AC squared equals 2 times AD squared plus 2 times BD squared. Plugging in the side lengths, we get 9 squared plus 13 squared equals 2 times AD squared plus 2 times 5 squared, which simplifies to 81 plus 169 equals 2 times AD squared plus 50. Solving for AD gives the square root of 100, which is 10. Since the centroid G divides the median AD in a 2:1 ratio, GD is one-third of AD, resulting in a length of 10 divided by 3.

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