Find the perimeter of the polygon [tex]$A B C D$[/tex] that has its vertices at [tex]$A(2,-2), B(6,-2)$[/tex], [tex]$C(6,6)$[/tex], and [tex]$D(2,6)$[/tex].
A) 24 units
B) 12 units
C) 64 units
D) 32 units
Answer (2)
Calculate the length of side A B using the distance formula and find A B = 4 .
Calculate the length of side BC using the distance formula and find BC = 8 .
Calculate the length of side C D using the distance formula and find C D = 4 .
Calculate the length of side D A using the distance formula and find D A = 8 .
Add the lengths of all four sides to find the perimeter: P = 4 + 8 + 4 + 8 = 24 units.
Explanation
Problem Analysis We are given the coordinates of the vertices of a polygon A BC D : A ( 2 , − 2 ) , B ( 6 , − 2 ) , C ( 6 , 6 ) , and D ( 2 , 6 ) . Our goal is to find the perimeter of this polygon.
Calculate Side Lengths To find the perimeter, we need to calculate the length of each side of the polygon and then add them up. We can use the distance formula to find the length of each side. The distance formula between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by:
( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
Let's calculate the length of each side:
Side A B : A ( 2 , − 2 ) and B ( 6 , − 2 )
A B = ( 6 − 2 ) 2 + ( − 2 − ( − 2 ) ) 2 = 4 2 + 0 2 = 16 = 4
Side BC : B ( 6 , − 2 ) and C ( 6 , 6 )
BC = ( 6 − 6 ) 2 + ( 6 − ( − 2 ) ) 2 = 0 2 + 8 2 = 64 = 8
Side C D : C ( 6 , 6 ) and D ( 2 , 6 )
C D = ( 2 − 6 ) 2 + ( 6 − 6 ) 2 = ( − 4 ) 2 + 0 2 = 16 = 4
Side D A : D ( 2 , 6 ) and A ( 2 , − 2 )
D A = ( 2 − 2 ) 2 + ( − 2 − 6 ) 2 = 0 2 + ( − 8 ) 2 = 64 = 8
Calculate Perimeter Now, we add the lengths of all four sides to find the perimeter:
P = A B + BC + C D + D A = 4 + 8 + 4 + 8 = 24
Therefore, the perimeter of the polygon A BC D is 24 units.
Final Answer The perimeter of the polygon A BC D is 24 units.
Examples
Understanding perimeters is crucial in many real-world applications. For instance, if you're planning to build a fence around a rectangular garden, knowing the perimeter helps you determine the amount of fencing material you need. Similarly, architects use perimeters to calculate the amount of material needed to frame windows or outline buildings. This concept is also fundamental in creating scale models, where maintaining accurate perimeters ensures the model accurately represents the original structure.
The perimeter of the polygon A BC D is 24 units, calculated by summing the lengths of its sides: 4 units (AB), 8 units (BC), 4 units (CD), and 8 units (DA). The total length is 4 + 8 + 4 + 8 = 24 units.
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