Chapter 13.5, Problem 14WE

a.

To determine

To show: that $\Delta RST$ is isosceles triangle

Explanation of Solution

Given information: The points are $R\left(1,0\right),S\left(7,4\right),\&T\left(11,-2\right)$

Proof:

  $\begin{array}{l}R{S}^2={\left(7-1\right)}^2+{\left(4-0\right)}^2\\ R{S}^2={\left(6\right)}^2+{\left(4\right)}^2\\ R{S}^2=36+16=52\\ RS=\sqrt{52}\\ S{T}^2={\left(11-7\right)}^2+{\left(-2-4\right)}^2\\ S{T}^2={\left(6\right)}^2+{\left(4\right)}^2\\ S{T}^2=36+16=52\\ ST=\sqrt{52}\\ RS=ST\end{array}$

Thus it is an isosceles triangle

b.

To determine

To find: the coordinate of $K$

Answer to Problem 14WE

The coordinate of $K$ is

Explanation of Solution

Given information: The altitude from the vertex meets the base at $K$

Calculation:

  $\begin{array}{l}K=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ K=\left(\frac{1+11}{2},\frac{0-2}{2}\right)\\ K=\left(\frac{12}{2},\frac{-2}{2}\right)\\ K=\left(6,-1\right)\end{array}$

Since, the altitude of an isosceles triangle meeting at base, divides the base into two equal parts.

So, $K$ is the midpoint