Question

first, find the length ( ell ) of the rectangle by using the distance formula. let the length be equal to ( ad ).
( ell=sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}=sqrt{1 - (-4)^{2}+(7 - 2)^{2}}=sqrt{+}=__ )
distance formula
let ( (x_{1},y_{1}) = a(-4,2) ) and ( (x_{2},y_{2}) = d(1,7) ).
subtract.
simplify.

next, find the width ( w ) of the rectangle by using the distance formula. let the width be equal to ( ab ).
( w=sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}=sqrt{0 - (-4)^{2}+(-2)-2^{2}}=sqrt{+}=__ )
distance formula
let ( (x_{1},y_{1}) = a(-4,2) ) and ( (x_{2},y_{2}) = b(0,-2) ).
subtract.
simplify.

use the length and width that you calculated to find the perimeter and area of the rectangle.
( p = 2ell+2w=2__ + 2sqrt{32}approx25.5 )
perimeter of a rectangle
( ell=sqrt{50} ) and ( w=sqrt{32} )
simplify.
the perimeter is about __ units.

( a=ell w=__\timessqrt{32}=40 )
area of a rectangle
( ell=sqrt{50} ) and ( w=sqrt{32} )
simplify.
the area is __ square units.

Explanation:

Step1: Calculate length $\ell$

Given $A(-4,2)$ and $D(1,7)$, using distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $\ell=\sqrt{[1-(-4)]^2+(7 - 2)^2}=\sqrt{5^2+5^2}=\sqrt{25 + 25}=\sqrt{50}$.

Step2: Calculate width $w$

Given $A(-4,2)$ and $B(0,-2)$, using distance formula, $w=\sqrt{[0-(-4)]^2+[(-2)-2]^2}=\sqrt{4^2+(-4)^2}=\sqrt{16 + 16}=\sqrt{32}$.

Step3: Calculate perimeter $P$

Using formula $P = 2\ell+2w$, substitute $\ell=\sqrt{50}$ and $w = \sqrt{32}$, $P=2\sqrt{50}+2\sqrt{32}\approx2\times7.07+2\times5.66=14.14 + 11.32\approx25.5$.

Step4: Calculate area $A$

Using formula $A=\ell w$, substitute $\ell=\sqrt{50}$ and $w=\sqrt{32}$, $A=\sqrt{50}\times\sqrt{32}=\sqrt{50\times32}=\sqrt{1600}=40$.

Answer:

The perimeter is about $25.5$ units.
The area is $40$ square units.