Chapter 4.8, Problem 23PPS

To determine

To prove:

The triangle formed by three cities is scalene triangle

Explanation of Solution

Given information:

Latitude and longitude Raleigh: $35.82°\mathrm{N},78.64°\mathrm{W}$

Latitude and longitude Durham: $35.99°\mathrm{N},78.91°\mathrm{W}$

Latitude and longitude Chapel Hill: $35.92°\mathrm{N},79.04°\mathrm{W}$

Proof:

Here, first draw triangle $\mathrm{\Delta }\mathrm{C}\mathrm{D}\mathrm{R}$

on a coordinate plane by using the latitude and longitude of given cities as:

  

Then need to estimate the side length of the triangle. If no two sides of $\mathrm{\Delta }\mathrm{C}\mathrm{D}\mathrm{R}$ are congruent, then the triangle formed by these three cities is scalene.

The distance formula is given by the following expression:

  $\mathrm{d}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}$

Use the distance formula to find the side lengths of $\mathrm{\Delta }\mathrm{C}\mathrm{D}\mathrm{R}$ :

  $\begin{array}{l}\mathrm{D}\mathrm{R}=\sqrt{{\left(35.99-35.82\right)}^2+{\left(78.91-78.64\right)}^2}\approx 0.32\\ \mathrm{C}\mathrm{R}=\sqrt{{\left(35.92-35.82\right)}^2+{\left(79.07-78.64\right)}^2}\approx 0.41\\ \mathrm{C}\mathrm{D}=\sqrt{{\left(35.99-35.92\right)}^2+{\left(78.91-79.04\right)}^2}\approx 0.15\end{array}$

Therefore, each side is of a different length in $\mathrm{\Delta }\mathrm{C}\mathrm{D}\mathrm{R}$ . Therefore, $\mathrm{\Delta }\mathrm{C}\mathrm{D}\mathrm{R}$ formed by latitude and longitude of given cities is scalene triangle.

Hence proved