#### Question

In ∆ABC, AB = 10, AC = 7, BC = 9 then find the length of the median drawn from point C to side AB

#### Solution

In ∆ACB, point D is the midpoint of side AB.

\[AD = BD = \frac{1}{2}AB = 5\]

\[{CA}^2 + {CB}^2 = 2 {DC}^2 + 2 {AD}^2 \left( \text{by Apollonius theorem} \right)\]

\[ \Rightarrow 7^2 + 9^2 = 2 {DC}^2 + 2\left( 5^2 \right)\]

\[ \Rightarrow 49 + 81 = 2 {DC}^2 + 2\left( 25 \right)\]

\[ \Rightarrow 130 = 2 {DC}^2 + 50\]

\[ \Rightarrow 2 {DC}^2 = 130 - 50\]

\[ \Rightarrow 2 {DC}^2 = 80\]

\[ \Rightarrow {DC}^2 = 40\]

\[ \Rightarrow DC = 2\sqrt{10}\]

Hence, the length of the median drawn from point C to side AB is \[2\sqrt{10}\]

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Solution In ∆Abc, Ab = 10, Ac = 7, Bc = 9 Then Find the Length of the Median Drawn from Point C to Side Ab Concept: Pythagoras Theorem.