Chapter 7.3, Problem 30WE

To determine

To express: The AB in terms of other lengths.

Explanation of Solution

Given information:

The two similar triangles.

  $\text{Δ}ABC\sim \text{Δ}DEF$

Property of proportions:

(1) If $\frac{a}{b}=\frac{c}{d}$ is equivalent to:

  $\begin{array}{l}ad=bc\dots \dots (1)\\ \frac{a}{c}=\frac{b}{d}\dots \dots (2)\\ \frac{b}{a}=\frac{d}{c}\dots \dots (3)\\ \frac{a+b}{b}=\frac{c+d}{d}\dots \dots (4)\end{array}$

(2) If $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots ,$ then $\frac{a+c+e+\dots }{b+d+f+\dots }=\frac{a}{b}=\dots \dots \dots (5)$

Now consider two similar triangles.

  $\text{Δ}ABC\sim \text{Δ}DEF$

Therefore,

  $\begin{array}{l}\frac{AB}{DE}=\frac{BC}{EF}\\ AB=\frac{BC}{EF}\times DE\end{array}$

And now use the equation $(5),$ as the perimeter of the triangles.

Thus,

  $\begin{array}{l}\frac{AB}{DE}=\frac{\text{Perimeter ofΔ}ABC}{\text{Perimeter ofΔ}DEF}\\ \frac{AB}{DE}=\frac{AB+BC+CA}{DE+EF+FD}\\ AB=\left(\frac{AB+BC+CA}{DE+EF+FD}\right)\times DE\end{array}$

Hence, these are the two possible answers.