Chapter 7.2, Problem 46PPS

To determine

To Prove: The given theorem.

Explanation of Solution

Given:

The give theorem is that $\frac{\text{perimeter}\text{of}\Delta ABC}{\text{perimeter}\text{of}\Delta DEF}=\frac{m}{n}$ .

The triangles ABC an DEF are congruent and $\frac{AB}{DE}=\frac{m}{n}$ .

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider that the two polygons are similar when the corresponding angles are similar and the corresponding side length are proportional.

As both the triangles are congruent. Then,

  $\begin{array}{c}\frac{AC}{DF}=\frac{BC}{EF}\\ =\frac{AB}{DE}\\ =\frac{m}{n}\end{array}$

Then, by the cross product property $AB$ is,

  $\begin{array}{l}AB=\frac{m}{n}\left(DE\right)\\ BC=\frac{m}{n}\left(EF\right)\\ AC=\frac{m}{n}\left(DF\right)\end{array}$

Then, use substitution to analyse the parameter as,

  $\begin{array}{c}{P}_{ABC}=AB+BC+AC\\ =DE\left(\frac{m}{n}\right)+EF\left(\frac{m}{n}\right)+DF\left(\frac{m}{n}\right)\\ =\frac{m}{n}\left(DE+EF+DF\right)\end{array}$

The ratio of the perimeter of the two triangles is,

  $\begin{array}{c}\frac{P_{\Delta ABC}}{P_{\Delta DEF}}=\frac{\frac{m}{n}\left(DE+EF+DF\right)}{\left(DE+EF+DF\right)}\\ =\frac{m}{n}\end{array}$

Hence, the formula is proved.