To determine the value of [tex]\( y \)[/tex] that ensures the pool labeled ABCD is a rectangle, we need to analyze the properties of a rectangle. One key property of a rectangle is that its diagonals are equal in length. Therefore, we need to set the lengths of the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal to each other and solve for [tex]\( y \)[/tex].
Here are the equations for the diagonals given in the problem:
[tex]\[ AC = 15y - 7 \][/tex]
[tex]\[ BD = 2y + 6 \][/tex]
Since for a rectangle, the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] must be equal, we set these two expressions equal to each other:
[tex]\[ 15y - 7 = 2y + 6 \][/tex]
To solve for [tex]\( y \)[/tex], we follow these steps:
1. Isolate the variable [tex]\( y \)[/tex] on one side of the equation. First, subtract [tex]\( 2y \)[/tex] from both sides:
[tex]\[ 15y - 2y - 7 = 6 \][/tex]
2. Simplify the equation:
[tex]\[ 13y - 7 = 6 \][/tex]
3. Add 7 to both sides to move the constant term away from the [tex]\( y \)[/tex] term:
[tex]\[ 13y = 13 \][/tex]
4. Divide both sides by 13 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{13}{13} \][/tex]
[tex]\[ y = 1 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is [tex]\( 1 \)[/tex].
The correct answer is:
[tex]\[ 1 \][/tex]
