To determine the value of [tex]\( y \)[/tex] that ensures the pool [tex]\( ABCD \)[/tex] is a rectangle, we start by analyzing the given conditions. For the pool to be a rectangle, the diagonals must be equal in length. In the given problem, the lengths of the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] are provided by the equations:
[tex]\[ AC = 10y + 4 \][/tex]
[tex]\[ BD = 13y - 8 \][/tex]
Since [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] should be equal for [tex]\( ABCD \)[/tex] to be a rectangle, we set these two expressions equal to each other and solve for [tex]\( y \)[/tex]:
[tex]\[ 10y + 4 = 13y - 8 \][/tex]
To isolate [tex]\( y \)[/tex], we first subtract [tex]\( 10y \)[/tex] from both sides:
[tex]\[ 4 = 13y - 10y - 8 \][/tex]
[tex]\[ 4 = 3y - 8 \][/tex]
Next, we add 8 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ 4 + 8 = 3y \][/tex]
[tex]\[ 12 = 3y \][/tex]
Finally, we divide both sides by 3:
[tex]\[ y = \frac{12}{3} \][/tex]
[tex]\[ y = 4 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is:
[tex]\[ y = 4 \][/tex]
