Chapter 13.7, Problem 34WE

a.

To determine

To find : the slope of diagonal $\overleftrightarrow{EK}$

Answer to Problem 34WE

The slope of diagonal $\overleftrightarrow{EK}$ is $\frac{1}{2}$

Explanation of Solution

Given information :

Quad. $BECK$ Is known to be a rhombus. Two of the vertices are $B\left(3,5\right)\text{ and }C\left(7,-3\right)$

Formula used :

  $m=\frac{y-y_1}{x-x_1}$

Calculation :

The slope of the line $BC$

  $\begin{array}{l}=m\\ =\frac{5-\left(-3\right)}{3-7}\\ =\frac{8}{-4}\\ =-2\end{array}$

Let $M$ be the point of intersect of $BC$ .

The line $BC$ is perpendicular to the line $EM+MK$ .

The required line has slope

  $\frac{-1}{m}=\frac{-1}{-2}=\frac{1}{2}$

Therefore, the slope of diagonal $\overleftrightarrow{EK}$ is $\frac{1}{2}$

Hence, the slope of diagonal $\overleftrightarrow{EK}$ is $\frac{1}{2}$ .

b.

To determine

To find : the equation of $\overleftrightarrow{EK}$

Answer to Problem 34WE

The equation of $\overleftrightarrow{EK}$ is $x-2y-3=0$

Explanation of Solution

Given information :

Quad. $BECK$ Is known to be a rhombus. Two of the vertices are $B\left(3,5\right)\text{ and }C\left(7,-3\right)$

Formula used :

  $y-y_1=m\left(x-x_1\right)$

Calculation :

The point of bisection

  $\begin{array}{l}=\left(\frac{3+7}{2},\frac{5-3}{2}\right)\\ =\left(5,1\right)\end{array}$

The slope of diagonal $\overleftrightarrow{EK}$ is $\frac{1}{2}$ .

Using $\left(5,1\right)$ , the equation is

  $\begin{array}{l}y-1=\frac{1}{2}\left(x-5\right)\\ y=\frac{1}{2}x-\frac{5}{2}+1\\ y=\frac{1}{2}x-\frac{3}{2}\\ 2y=x-3\\ x-2y-3=0\end{array}$

Hence, the equation of $\overleftrightarrow{EK}$ is $x-2y-3=0$ .