A contractor is building a pool labeled [tex]\(ABCD\)[/tex] on the plans. If [tex]\(AC = 15y - 7\)[/tex] and [tex]\(BD = 2y + 6\)[/tex], what value of [tex]\(y\)[/tex] ensures the pool is a rectangle?

A. [tex]\(-13\)[/tex]
B. [tex]\(-1\)[/tex]
C. [tex]\(13\)[/tex]



Answer :

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.