To determine the value of [tex]\( y \)[/tex] that ensures the pool labeled [tex]\( ABCD \)[/tex] is a rectangle, we need to use the property that the diagonals of a rectangle are equal in length.
Given:
[tex]\[ AC = 15y - 7 \][/tex]
[tex]\[ BD = 2y + 6 \][/tex]
Since the diagonals of a rectangle are equal, we equate [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex]:
[tex]\[ 15y - 7 = 2y + 6 \][/tex]
Next, we solve this equation for [tex]\( y \)[/tex].
First, we combine like terms by subtracting [tex]\( 2y \)[/tex] from both sides:
[tex]\[ 15y - 2y - 7 = 6 \][/tex]
[tex]\[ 13y - 7 = 6 \][/tex]
Next, we isolate the term involving [tex]\( y \)[/tex] by adding 7 to both sides:
[tex]\[ 13y - 7 + 7 = 6 + 7 \][/tex]
[tex]\[ 13y = 13 \][/tex]
Finally, we solve for [tex]\( y \)[/tex] by dividing both sides by 13:
[tex]\[ y = \frac{13}{13} \][/tex]
[tex]\[ y = 1 \][/tex]
Thus, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is:
[tex]\[ y = 1 \][/tex]
However, [tex]\( y = 1 \)[/tex] is not among the provided options. Thus, examining the given answer choices:
[tex]\[
\begin{array}{ccc}
-13 & -1 & 13 \\
\end{array}
\][/tex]
None of these values for [tex]\( y \)[/tex] satisfies the requirement [tex]\( y = 1 \)[/tex], so it appears there may have been a mistake in the options provided.
However, since the question asks for an eligible value, and the provided answer should be thus amongst the options, a discrepancy in the scenario has arisen - suggesting a likely mistake in given alternatives or in the theoretical assumption.
Since none accurately satisfies [tex]\( y = 1 \)[/tex] within provided options here, question adjustment or potential confirmation might be prudent.
