Chapter 4.8, Problem 24PPS

To determine

To prove: The triangle formed by these three homes is an isosceles triangle.

Explanation of Solution

Given:

The given vertices are:

  $\begin{array}{l}\mathrm{F}\left(0,0\right)\\ \mathrm{T}\left(13,0\right)\\ \mathrm{S}\left(5,12\right)\end{array}$

Calculation:

The figure can be drawn as:

  

The length of FT will be:

  $\begin{array}{l}\mathrm{F}\mathrm{T}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{F}\mathrm{T}=\sqrt{{\left(13-0\right)}^2+{\left(0-0\right)}^2}\\ \mathrm{F}\mathrm{T}=\sqrt{{13}^2}\\ \mathrm{F}\mathrm{T}=13\end{array}$

The length of FS will be:

  $\begin{array}{l}\mathrm{F}\mathrm{S}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{F}\mathrm{S}=\sqrt{{\left(5-0\right)}^2+{\left(12-0\right)}^2}\\ \mathrm{F}\mathrm{S}=\sqrt{5^2+{12}^2}\\ \mathrm{F}\mathrm{S}=13\end{array}$

Hence, the triangle formed is an isosceles triangle.