Question

Consider a line segment with endpoints (4, 7) and (1, 11) Which line segment is equal in distance to the given line segment?

A. line segment with endpoints (−6, 4) and (2, −5).
B. line segment with endpoints (1, −4) and (9, 2).
C. line segment with endpoints (−3, 1) and (4, 5).
D.A line segment with endpoints (5, 3) and (1, 6).

Answer 1

well, first off, let's see how long the segment from (4 , 7) to (1 , 11) is

`~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{4}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{11})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ d=√((~~1 - 4~~)^2 + (~~11 - 7~~)^2)\implies d=√( (-3)^2 + (4)^2) \\\\\\ d=√( 9 + 16)\implies d=√( 25 )\implies d=5`

now let's check the other points

`~\hfill \stackrel{\textit{\large distance between 2 points}}{d = √(( x_2- x_1)^2 + ( y_2- y_1)^2)}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ (\stackrel{x_1}{-6}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{-5}) ~\hfill d=√((~~ 2- (-6)~~)^2 + (~~ -5- 4~~)^2) \\\\\\ ~\hfill d=√(( 8 )^2 + ( -9)^2) \implies d=√( 145) \\\\[-0.35em] ~\dotfill`

`(\stackrel{x_1}{1}~,~\stackrel{y_1}{-4})\qquad (\stackrel{x_2}{9}~,~\stackrel{y_2}{2}) ~\hfill d=√((~~ 9- 1~~)^2 + (~~ 2- (-4)~~)^2) \\\\\\ ~\hfill d=√(( 8)^2 + ( 6)^2) \implies d=√( 100)\implies d=10 \\\\[-0.35em] ~\dotfill\\\\ (\stackrel{x_1}{-3}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{5}) ~\hfill d=√((~~ 4- (-3)~~)^2 + (~~ 5- 1~~)^2) \\\\\\ ~\hfill d=√(( 7 )^2 + ( 4)^2) \implies d=√( 65) \\\\[-0.35em] ~\dotfill`

`(\stackrel{x_1}{5}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{6}) ~\hfill d=√((~~ 1- 5~~)^2 + (~~ 6- 3~~)^2) \\\\\\ ~\hfill d=√(( -4)^2 + ( 3)^2) \implies d=√( 25)\implies d=5\textit{\LARGE \checkmark}`