To determine which of the given points lie on the line that represents the relationship between the side length of a square, [tex]\( x \)[/tex], and its perimeter, [tex]\( y \)[/tex], we first need to establish the relationship itself.
For a square, the perimeter is calculated by multiplying the side length by 4. Mathematically, this relationship can be expressed as:
[tex]\[ y = 4x \][/tex]
We will now check each of the given points to see if they satisfy this equation.
Point A: (2, 6)
1. Substitute [tex]\( x = 2 \)[/tex] into the equation [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(2) \][/tex]
[tex]\[ y = 8 \][/tex]
2. Here, [tex]\( y = 6 \)[/tex], which does not satisfy the equation [tex]\( y = 8 \)[/tex].
Therefore, the point (2, 6) does not lie on the line.
Point B: (2, 8)
1. Substitute [tex]\( x = 2 \)[/tex] into the equation [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(2) \][/tex]
[tex]\[ y = 8 \][/tex]
2. Here, [tex]\( y = 8 \)[/tex], which satisfies the equation [tex]\( y = 8 \)[/tex].
Therefore, the point (2, 8) lies on the line.
Point C: (6, 2)
1. Substitute [tex]\( x = 6 \)[/tex] into the equation [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(6) \][/tex]
[tex]\[ y = 24 \][/tex]
2. Here, [tex]\( y = 2 \)[/tex], which does not satisfy the equation [tex]\( y = 24 \)[/tex].
Therefore, the point (6, 2) does not lie on the line.
Point D: (8, 2)
1. Substitute [tex]\( x = 8 \)[/tex] into the equation [tex]\( y = 4x \)[/tex]:
[tex]\[ y = 4(8) \][/tex]
[tex]\[ y = 32 \][/tex]
2. Here, [tex]\( y = 2 \)[/tex], which does not satisfy the equation [tex]\( y = 32 \)[/tex].
Therefore, the point (8, 2) does not lie on the line.
Based on this analysis, the point with coordinates [tex]\((2, 8)\)[/tex] (Option B) is the one that lies on the line representing the relationship between the side length of a square and its perimeter.
