Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Check the picture below.
[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ Q(\stackrel{x_1}{-4}~,~\stackrel{y_1}{9})\qquad U(\stackrel{x_2}{2}~,~\stackrel{y_2}{3}) ~\hfill QU=\sqrt{[ 2- (-4)]^2 + [ 3- 9]^2} \\\\\\ ~\hfill \boxed{QU=\sqrt{72}} \\\\\\ U(\stackrel{x_1}{2}~,~\stackrel{y_1}{3})\qquad A(\stackrel{x_2}{-3}~,~\stackrel{y_2}{-2}) ~\hfill UA=\sqrt{[ -3- 2]^2 + [ -2- 3]^2} \\\\\\ ~\hfill \boxed{UA=\sqrt{50}}[/tex]
[tex]A(\stackrel{x_1}{-3}~,~\stackrel{y_1}{-2})\qquad D(\stackrel{x_2}{-9}~,~\stackrel{y_2}{4}) ~\hfill AD=\sqrt{[ -9- (-3)]^2 + [ 4- (-2)]^2} \\\\\\ ~\hfill \boxed{AD=\sqrt{72}} \\\\\\ D(\stackrel{x_1}{-9}~,~\stackrel{y_1}{4})\qquad Q(\stackrel{x_2}{-4}~,~\stackrel{y_2}{9}) ~\hfill DQ=\sqrt{[ -4- (-9)]^2 + [ 9- 4]^2} \\\\\\ ~\hfill \boxed{DQ=\sqrt{50}}[/tex]
now, let's take a peek at that above, DQ = UA and QU = AD, so opposite sides of the polygon are equal.
Now, let's check the slopes of DQ and QU
[tex]D(\stackrel{x_1}{-9}~,~\stackrel{y_1}{4})\qquad Q(\stackrel{x_2}{-4}~,~\stackrel{y_2}{9}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{9}-\stackrel{y1}{4}}}{\underset{run} {\underset{x_2}{-4}-\underset{x_1}{(-9)}}} \implies \cfrac{9 -4}{-4 +9}\implies \cfrac{5}{5}\implies \cfrac{1}{1}\implies 1 \\\\[-0.35em] ~\dotfill[/tex]
[tex]Q(\stackrel{x_1}{-4}~,~\stackrel{y_1}{9})\qquad U(\stackrel{x_2}{2}~,~\stackrel{y_2}{3}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{3}-\stackrel{y1}{9}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{(-4)}}} \implies \cfrac{3 -9}{2 +4}\implies \cfrac{-6}{6}\implies \cfrac{-1}{1}\implies -1[/tex]
keeping in mind that perpendicular lines have negative reciprocal slopes, let's notice that the reciprocal of 1/1 is just 1/1 and the negative of that is just -1/1 or -1, so QU has a slope that is really just the negative reciprocal of DQ, those two lines are perpendicular, thus making a 90° angle, and their congruent opposite sides will also make a 90°, that makes QUAD hmmm yeap, you guessed it.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
