Answer:
Ratio of Area of Rectangle ABEF to Rectangle ACDF is 2 : 3
Area of rectangle ABEF is 10√41 units²
Perimeter of rectangle BCDE is ( 10 + 2√41 ) units²
Step-by-step explanation:
Given: ABEF and BCDE and ACDF are rectangles.
coordinates of F( 5 , 2 ) , E( 11 , 10 ) , D( 14 , 14 ) and A( 0 , 6 )
To find: Ratio of Area of Rectangle ABEF to Rectangle ACDF
Area of rectangle ABEF
Perimeter of BCDE.
We know that, Area of Rectangle = Length × Width
In rectangle ABEF
length = FE = [tex]\sqrt{(11-5)^2+(10-2)^2}=\sqrt{6^2+8^2}=\sqrt{36+64}=10\,units[/tex]
Width = FA
Area of Rectangle ABEF = FE × FA = ( 10 × FA ) units²
In rectangle ACDF
length = FD = [tex]\sqrt{(14-5)^2+(14-2)^2}=\sqrt{9^2+12^2}=\sqrt{81+144}=15\,units[/tex]
Width = FA
Area of Rectangle ACDF = FD × FA = ( 15 × FA ) units²
[tex]\frac{Area\:of\:Rectangle\:ABEF }{Area\:of\:Rectangle\:ACDF}=\frac{10\times FA}{15\times FA}=\frac{2}{3}[/tex]
Thus, Ratio of Area of Rectangle ABEF to Rectangle ACDF is 2 : 3
In rectangle ABEF
length = FE = 10 units (from above)
Width = FA = [tex]\sqrt{(0-5)^2+(6-2)^2}=\sqrt{(-5)^2+4^2}=\sqrt{25+16}=\sqrt{41}\,units[/tex]
Area of Rectangle ABEF = FE × FA = 10 × √41 = 10√41 units²
Thus, Area of rectangle ABEF is 10√41 units²
In rectangle BCDE
length = DE = [tex]\sqrt{(14-11)^2+(14-10)^2}=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\,units[/tex]
Width = CD = FA = √41 units
Perimeter of Rectangle BCDE = 2 × ( DE + CD ) = 2 × (5 + √41) = ( 10 + 2√41 ) units²
Thus, Perimeter of rectangle BCDE is ( 10 + 2√41 ) units²
