Question

Miguel plots points A, B, and C in the coordinate plane.

A. What is the distance between points A and B? Explain your reasoning.
B. Is (gif triangle) ABC an equilateral triangle? Explain your reasoning.

Answer 1

is △ABC equilateral? well, we dunno, however let's check for all sides' length.

`~\hfill \stackrel{\textit{\large distance between 2 points}}{d = √(( x_2- x_1)^2 + ( y_2- y_1)^2)}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ A(\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad B(\stackrel{x_2}{-6}~,~\stackrel{y_2}{-2}) ~\hfill AB=√((~~ -6- (-3)~~)^2 + (~~ -2- 2~~)^2) \\\\\\ ~\hfill AB=√(( -3)^2 + ( -4)^2) \implies AB=√( 25 )\implies \boxed{AB=5}`

`B(\stackrel{x_1}{-6}~,~\stackrel{y_1}{-2})\qquad C(\stackrel{x_2}{0}~,~\stackrel{y_2}{-2}) ~\hfill BC=√((~~ 0- (-6)~~)^2 + (~~ -2- (-2)~~)^2) \\\\\\ ~\hfill BC=√(( 6)^2 + ( 0)^2) \implies BC=√( 36 )\implies \boxed{BC=6} \\\\\\ C(\stackrel{x_1}{0}~,~\stackrel{y_1}{-2})\qquad A(\stackrel{x_2}{-3}~,~\stackrel{y_2}{2}) ~\hfill CA=√((~~ -3- 0~~)^2 + (~~ 2- (-2)~~)^2) \\\\\\ ~\hfill CA=√(( -3)^2 + (4)^2) \implies CA=√( 25 )\implies \boxed{CA=5}`

well, not quite.