Chapter 4, Problem 41SGR

To determine

The coordinate proof to show that the triangle formed is Scalene.

Explanation of Solution

Given:

The given triangle is:

  

Calculation:

The triangle can be drawn as:

  

The length of the sides will be:

  $\begin{array}{c}SD=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(8-0\right)}^2+{\left(28-0\right)}^2}\\ =\sqrt{848}\\ \\ SH=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(19-0\right)}^2+{\left(7-0\right)}^2}\\ =410\\ \\ DH=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(19-8\right)}^2+{\left(7-28\right)}^2}\\ =\sqrt{562}\end{array}$

No two sides of triangle are equal; hence, the triangle is Scalene.