Chapter 9.3, Problem 76SR

To determine

To Find:The midpoints of the given end points and then find the distance between them.

Answer to Problem 76SR

  $d=\sqrt{271}$

Explanation of Solution

Given:End points are $\left(2\sqrt{3},-5\right),\left(-3\sqrt{3},9\right)$

Using the Mid-Point formula

  $(h,k)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Here

  $\begin{array}{l}{x}_1=2\sqrt{3}\\ y_1=-5\\ x_2=-3\sqrt{3}\\ y_2=9\end{array}$

Now

  $\begin{array}{l}(h,k)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ (h,k)=\left(\frac{2\sqrt{3}-3\sqrt{3}}{2},\frac{-5+9}{2}\right)\\ (h,k)=\left(\frac{-\sqrt{3}}{2},2\right)\end{array}$

Now

Using the distance formula

  $d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

  $\begin{array}{l}d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}\\ d=\sqrt{{\left(-3\sqrt{3}-2\sqrt{3}\right)}^2+{\left(9+5\right)}^2}\\ d=\sqrt{75+196}\\ d=\sqrt{271}\end{array}$

Hence, $d=\sqrt{271}$