Chapter 5.5, Problem 17PSB

To determine

To prove: $\Delta ARS$ is isosceles

Explanation of Solution

Given information:

  $VRST$ is an isosceles trapezoid with legs $\overline{VR}\text{ and }\overline{\text{TS}}$ .

Formula used:

In isosceles trapezoid $VRST$

• Diagonals are congruent.
• The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, i.e., $\frac{RA}{TA}=\frac{SA}{VA}=\frac{RS}{TV}$

Proof:

To prove $\Delta ARS$ is isosceles, it is enough to prove $SA=RA$ .

From the properties of an isosceles trapezoid:

Diagonals are congruent i.e, $VS=RT$

  $\Rightarrow VA+SA=RA+TA\dots \left(i\right)$

The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect i.e., $\frac{RA}{TA}=\frac{SA}{VA}=\frac{l}{m}$

  $\begin{array}{l}\Rightarrow \frac{RA}{TA}=\frac{l}{m}\text{ and }\frac{SA}{VA}=\frac{l}{m}\\ \Rightarrow TA=\frac{m}{l}RA\text{ and }VA\text{=}\frac{m}{l}SA\dots \left(ii\right)\end{array}$

Simplifying $\left(i\right)\text{ and }\left(ii\right)$ , we get

  $\begin{array}{c}\left(i\right)\Rightarrow VA+SA=RA+TA\\ \Rightarrow \frac{m}{l}SA+SA=RA+\frac{m}{l}RA\\ \Rightarrow \left(\frac{m}{l}+1\right)SA=\left(\frac{m}{l}+1\right)RA\\ \Rightarrow SA=RA\end{array}$

Hence, $\Delta ARS$ is an isosceles triangle.