Chapter 12.8, Problem 19E

To determine

To Prove: The triangle formed by the position, your position and the campsite is isosceles.

Explanation of Solution

Given:

The hike is done to point of 500 m east and 1200 m north of the campsite.

The cousin hike to the points that is 1000 m east of the campsite.

Calculation:

The value of the length of $AB$ is,

  $\begin{array}{c}AB=\sqrt{{\left(1000-500\right)}^2+{\left(0-1200\right)}^2}\end{array}$

    $\begin{array}{c}=1300\end{array}$

The value of the length of $BO$ is,

  $\begin{array}{c}BO=\sqrt{{\left(0-1000\right)}^2+{\left(0-0\right)}^2}\end{array}$

    $\begin{array}{c}=1000\end{array}$

The value of the length of $AO$ is,

  $\begin{array}{c}AO=\sqrt{{\left(0-500\right)}^2+{\left(0-1200\right)}^2}\end{array}$

    $\begin{array}{c}=1300\end{array}$

This shows that $AB=AO$ and by the definition congruent segments $\overline{AB}\cong \overline{AO}$ this shows that the triangle is isosceles triangle.

Thus, the triangle formed by the positions and the campsite is isosceles.