A line segment has endpoints at (1 ,9 ) and (6 ,7 ). The line segment is dilated by a factor of 4  around (4 ,3 ). What are the new endpoints and length of the line segment?

Nov 2, 2016

$\text{the new end points of line segment are:(-8,27),(12,19)}$
$\text{the length of line segment is :} \sqrt{464}$

Explanation:

$\text{the point of D(4,3) is dilatation point}$

$\text{the point F is image of B}$

$\text{distance between BC is :} 7 - 3 = 4$

$\text{distance between FG is :"4*"factor} = 4 \cdot 4 = 16$

$\text{y coordinate of F is :} 3 + 16 = 19$

$\text{distance between DC is :} 6 - 4 = 2$

$\text{distance between DG is :"2*"factor} = 2 \cdot 4 = 8$

$\text{x coordinate of F is :} 4 + 8 = 12$

$\text{coordinates of F are :} F \left(12 , 19\right)$

$\text{The point of E is image of A}$

$\text{distance between AH is:} 9 - 3 = 6$

$\text{distance between EI is:"6*"factor} = 6 \cdot 4 = 24$

$\text{y coordinate of E is :} 3 + 24 = 27$

$\text{distance between HD is :} 4 - 1 = 3$

$\text{distance between ID is :"3*"factor} = 3 \cdot 4 = 12$

$\text{the x coordinate of E is :} 4 - 12 = - 8$

$\text{coordinates of E are :} E \left(- 8 , 27\right)$

$\text{length of line segment AB :}$

$\overline{A B} = \sqrt{{\left(6 - 1\right)}^{2} + {\left(7 - 9\right)}^{2}}$

$\overline{A B} = \sqrt{25 + 4} = \sqrt{29}$

$\text{length of line segment EF:}$

$\overline{E F} = \sqrt{{\left(12 + 8\right)}^{2}} + {\left(19 - 27\right)}^{2}$

$\overline{E F} = \sqrt{400 + 64} = \sqrt{464}$