Given that our equation is in slop-intercept form

\(\displaystyle y=mx+b\)

\(\displaystyle y=\frac{5}{9}x+8\)

where \(\displaystyle m\) is the slope, we see that for this line the slope is \(\displaystyle \frac{5}{9}\).

Find the negative reciprocal of the slope to find the perpendicular line's slope.

Flip the fraction and make it negative. 

\(\displaystyle \frac{5}{9}\rightarrow -\frac{9}{5}\).

Perpendicular lines are lines that intersect each other at a ninety degree angle. The slope of a line that is perpendicular to another has the opposite sign and is the reciprocal. The slope of a line perpendicular to the one given would be \(\displaystyle 2\).

\(\displaystyle \textup{Of the other three sides, the opposite side lays on a parallel line,}\\\textup{ so its slope is 3 as well. The two adjacent sides lay on perpendicular lines,}\\\textup{ so their slope is the inverse reciprocal of 3; that is, } -\frac{1}{3}.\)

 \(\displaystyle \textup{The segment between the two opposite vertices (0,0) and (-2,4) is one }\\\textup{diagonal of the square with slope}\frac{4}{-2}=-2 \textup{ and midpoint } \left(\frac{0+(-2)}{2},\frac{0+4}{2} \right)=(-1,2). \textup{ The other diagonal lays on the line } y=\frac{1}{2}x+p \textup{ since it is perpendicular}\\\textup{ and has as a slope of the inverse reciprocal. Substituting the midpoint into the }\\\textup{ parametric equation we obtain } p=\frac{5}{2}\textup{ and the line equation } y=\frac{1}{2}c+\frac{5}{2}.\)

  \(\displaystyle \textup{ Only point } (1,3) \textup{ among the possible answers lays on this line.}\). 

\(\displaystyle \textup{ Alternatively, one can check for every possible answer if the two segments}\\\textup{between the candidate point and the two opposite vertices are perpendicular.}\)

The slope of a perpendicular line is always the negative reciprocal of the original slope.

Our original equation is \(\displaystyle y=3x+4\) which is in the form \(\displaystyle y=mx+b\) where \(\displaystyle m\) is the slope. Therefore, the original slope is \(\displaystyle m=3\).

To find the slope of a line perpendicular we want to use the following formula,

\(\displaystyle m_p=-\frac{1}{m}\)

\(\displaystyle m_p=-\frac{1}{3}\).

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle \frac{1}{3}\).

\(\displaystyle \frac{1}{3}\rightarrow \frac{3}{1}=3\)

Next, change the sign.

\(\displaystyle 3\rightarrow-3\)

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle 2\):  

\(\displaystyle 2=\frac{2}{1}\)

Flip the numerator and the denominator.

\(\displaystyle \frac{2}{1}\rightarrow \frac{1}{2}\)

Next, change the sign.

\(\displaystyle \frac{1}{2}\rightarrow-\frac{1}{2}\)

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle -\frac{2}{3}\).

\(\displaystyle -\frac{2}{3}\rightarrow-\frac{3}{2}\)

Next, change the sign.

\(\displaystyle -\frac{3}{2}\rightarrow\frac{3}{2}\)

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle -10\).

\(\displaystyle -10=-\frac{10}{1}\)

Flip the numerator and the denominator.

\(\displaystyle -\frac{10}{1}=-\frac{1}{10}\)

Next, change the sign.

\(\displaystyle -\frac{1}{10}\rightarrow\frac{1}{10}\)

Lines can be written in the slope-intercept form: 

\(\displaystyle y=mx+b\)

In this equation, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Lines that are perpendicular to each other have slopes that are negative reciprocals of each other. This means that you need to flip the numerator and denominator of the given slope and then change the sign.

First, find the reciprocal of \(\displaystyle \frac{1}{4}\).

\(\displaystyle \frac{1}{4}\rightarrow \frac{4}{1}=4\)

Next, change the sign.

\(\displaystyle 4\rightarrow-4\)