Chapter 15, Problem 20RP

To determine

To Justify: The median of triangle is half of the sum of its adjacent sides.

Explanation of Solution

Given:

  $\begin{array}{l}In\Delta ABC\\ AM\text{ is median on }BC\\ BM=MC\end{array}$

Formula Used:

Triangle-Inequality Theorem: It states that sum of two sides of a triangle is always greater than the third.

SAS Congruence: It states that two triangle will be congruent if two adjacent sides and joining angle are congruent to other triangle.

Calculation:

  

  $\begin{array}{l}In\Delta ABC\\ AM\text{ is median on }BC\\ BM=MC\\ \text{Extend }AM\text{ to }Ps\text{uch that}\overline{AM}\cong \overline{MP}\\ \text{Join to form quadrilateral }ABPC\\ AM=MP\\ MC=BM\\ \angle AMC=\angle BMP\left(\text{Vertically opposite angle}\right)\\ \Delta AMC\cong \Delta BMP\left(SAS\text{ Congruence Rule}\right)\\ AP<AB+BP\\ AM+MP<AB+AC\\ \Rightarrow 2AM<AB+AC\\ \Rightarrow AM<\frac{1}{2}\left(AB+BC\right)\end{array}$

Conclusion:

Hence, Proved.