Multiple choice

A triangle ABC has lengths AB = 12cm and AC = 15 cm while BC = 18cm , then find the length of median AD in cm ?

  1. √103.5

  2. √100.5

  3. √90.5

  4. √84.5
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A Correct answer
Explanation

Using Apollonius theorem: AB^2 + AC^2 = 2(AD^2 + BD^2). Here BD = BC/2 = 9. 12^2 + 15^2 = 2(AD^2 + 9^2). 144 + 225 = 2(AD^2 + 81). 369 = 2AD^2 + 162. 207 = 2AD^2. AD^2 = 103.5. AD = sqrt(103.5).

AI explanation

Using Apollonius's theorem for the median of a triangle, the formula is AB^2 + AC^2 = 2(AD^2 + BD^2). With AB = 12, AC = 15, and BD = 9, the equation is 12^2 + 15^2 = 2(AD^2 + 9^2), simplifying to 369 = 2(AD^2 + 81). Solving for AD^2 gives 184.5 - 81 = 103.5, so the median AD is the square root of 103.5.

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