Question

Is her assertion correct ?

Answer 1

Check the picture below.

so, the vertex at N, is noticeably not a right angle is an acute angle, so is less than 90°, so we don't need to check that one.

now, is the angle at L 90°?

well, if that's true LM and LN are perpendicular, and if they're indeed perpendicular, their slopes are negative reciprocal, meaning the slope of one is the same as the other but negative and upside down, well, let's check.

`\bf L(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad M(\stackrel{x_2}{2}~,~\stackrel{y_2}{2}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{2-0}{2-0}\implies \cfrac{2}{2}\implies 1 \\\\[-0.35em] ~\dotfill\\\\ L(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad N(\stackrel{x_2}{2}~,~\stackrel{y_2}{-1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-1-0}{2-0}\implies \cfrac{-1}{2}\implies -\cfrac{1}{2}`

`\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope~of~LM}{1\implies \cfrac{1}{\underline{1}}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{\underline{1}}{1}}\qquad \stackrel{negative~reciprocal}{-\cfrac{\underline{1}}{1}\implies -1}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{slope of LM}}{1}\qquad \stackrel{\textit{negative reciprocal of LM}}{-1}\qquad \stackrel{\textit{slope of LN}}{-\cfrac{1}{2}}~\hfill -1\\e -\cfrac{1}{2}`

so that means Lydia put too much espresso on her last cup.