Chapter 5.1, Problem 14PSC

To determine

To Prove:

If no two medians of a triangle are congruent, then the triangle is scalene. .

Explanation of Solution

Given:

  

  $\text{A}\Delta ABC\text{with }\overline{AD}\text{ and }\overline{BE}\text{ as median such that}\overline{AD}\cancel{\cong }\overline{BE}.$

Concept Used:

Median is a line drawn from the vertex of a triangle on its opposite side divide its into two equal parts. Unequal medians of a triangle divides the sides of triangle into unequal parts..

Proof:

Here, we have $\overline{AD}$ is a median of side $\overline{BC}$

  $\Rightarrow \overline{BD}\cong \overline{CD}$

Also, $\overline{BE}$ is a median of side $\overline{AC}$

  $\Rightarrow \overline{AE}\cong \overline{CE}$

  $\begin{array}{l}\because \overline{AD}\cancel{\cong }\overline{BE}\\ \Rightarrow \frac{1}{2}l\left(\overline{BC}\right)\cancel{\cong }\frac{1}{2}l\left(\overline{AC}\right)\\ \Rightarrow \overline{BC}\cancel{\cong }\overline{AC}\end{array}$

  $\Delta ABC\text{is an scalene triangle}\text{.}$