Question

. Is the line through points P(–5, 10) and Q(–9, 2) perpendicular to the line through points R(4, 6) and S(–4, 10)? Explain. A.Yes; their slopes have product –1. B.No; their slopes are not equal. C.No, their slopes are not opposite reciprocals. D.Yes; their slopes are equal.

Answer 1

if two lines are perpendicular to each other, the product of their slope is -1.

so,... .hmmm let's check the slope of each one, and their product then.

$\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ -5}}\quad ,&{{ 10}})\quad % (c,d) &({{ -9}}\quad ,&{{ 2}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{2-10}{-9-(-5)}\implies \cfrac{2-10}{-9+5} \\\\\\ \cfrac{-8}{-4}\implies \stackrel{\textit{slope for PQ}}{2}\\\\ -------------------------------$

$\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 4}}\quad ,&{{ 6}})\quad % (c,d) &({{ -4}}\quad ,&{{ 10}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{10-6}{-4-4}\implies \cfrac{4}{-8} \\\\\\ \stackrel{\textit{slope for RS}}{-\cfrac{1}{2}}\\\\ -------------------------------\\\\ \stackrel{mPQ}{2}\cdot \stackrel{mRS}{-\cfrac{1}{2}}\implies -\cfrac{2}{2}\implies -1$