Question

Find \[AD\] with the help of Apollonius theorem, where $ \Delta ABC $ is equilateral with side $ = 5cm $ .

Answer 1

Hint: To solve this problem we should know about the equilateral triangle and about the Apollonius theorem.
Equilateral triangle: The triangle having all sides are equal and all angles between the sides are equal.
Apollonius theorem: It states that the sum of the square of any two sides of any triangle equal twice the square on half the third side, together with twice the square on the median bisecting the third side.

Complete step by step solution:
As we have given in the question,

Here, $ AB = 5cm $ .
As it is an equilateral triangle. So all sides will be equal.
As by using Apollonius theorem. We can write it in mathematical form,
  $ A{B^2} + A{C^2} = 2(A{D^2} + D{C^2}) $ ……………………… $ (1) $
As it is an equilateral triangle, $ AD $ bisect the base line $ BC $ .
So, $ DC = \dfrac{{BC}}{2} = \dfrac{5}{2} = 2.5cm $ .
Keeping values in $ (1) $ . We get,
  $ \Rightarrow {5^2} + {5^2} = 2(A{D^2} + {2.5^2}) $
Further solving it. We get,
 $ \Rightarrow \dfrac{{50}}{2} = A{D^2} + 6.25 $
Moving numerical term to one side,
 $ \Rightarrow 25 - 6.25 = A{D^2} $
 $ \Rightarrow 18.75 = A{D^2} $
By taking under-root on both side,
   $ \Rightarrow \sqrt {18.75} = AD = 4.33 cm $
Hence, $ AD = 4.33cm $
So, the correct answer is “ $ AD = 4.33 cm $ ”.

Note: We can also solve this problem by simply using Pythagoras theorem. As it is an equilateral triangle the median will bisect the base line in the right angle. So, the equilateral triangle will change into a right-angle triangle and by using Pythagoras triangle we can easily calculate it.