# A line segment has endpoints at (1 ,6 ) and (5 ,2 ). The line segment is dilated by a factor of 4  around (2 ,1 ). What are the new endpoints and length of the line segment?

Nov 25, 2016

$A ' \left(6 , - 19\right)$ and B'(-10;-3)

#### Explanation:

We have to solve the dilation equation of factor 4 for x and y

${x}_{C} = \frac{4 \cdot {x}_{A} + {x}_{A} '}{5}$ and ${y}_{C} = \frac{4 \cdot {y}_{A} + {y}_{A} '}{5}$

with C(2;1) and A(1;6) we have
$2 = \frac{4 \cdot 1 + {x}_{A} '}{5}$ from which it is ${x}_{A} ' = 6$

and $1 = \frac{4 \cdot 6 + {y}_{A} '}{5}$ from which it is ${y}_{B} ' = - 19$

the same for $B '$
$2 = \frac{4 \cdot 5 + {x}_{B} '}{5}$ from which it is ${x}_{B} ' = - 10$
$1 = \frac{4 \cdot 2 + {y}_{B} '}{5}$ from which it is ${y}_{B} ' = - 3$

It is enough to check that $A B = \sqrt[2]{{\left(5 - 1\right)}^{2} + {\left(2 - 6\right)}^{2}} = 4 \sqrt[2]{2}$
and $A ' B ' = \sqrt[2]{{\left(- 10 - 6\right)}^{2} + {\left(- 3 + 19\right)}^{2}} = 16 \sqrt[2]{2}$
$A ' B ' = 4 A B$