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A contractor is building a pool labeled [tex]$ABCD$[/tex] on the plans. If [tex]$AC = 10y + 4$[/tex] and [tex]$BD = 13y - 8$[/tex], what value of [tex]$y$[/tex] ensures the pool is a rectangle?

A. [tex]$-4$[/tex]
B. 4
C. [tex]$-12$[/tex]
D. 12

Sagot :

GinnyAnsweridn
To ensure that the pool is a rectangle, the diagonals of the rectangle, [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex], must be equal. This means we need to solve the equation [tex]\( AC = BD \)[/tex].

Given:
[tex]\[ AC = 10y + 4 \][/tex]
[tex]\[ BD = 13y - 8 \][/tex]

We set the expressions for [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal to each other:
[tex]\[ 10y + 4 = 13y - 8 \][/tex]

Now, solve for [tex]\( y \)[/tex]:

1. Subtract [tex]\( 10y \)[/tex] from both sides:
[tex]\[ 4 = 3y - 8 \][/tex]

2. Add 8 to both sides:
[tex]\[ 12 = 3y \][/tex]

3. Divide both sides by 3:
[tex]\[ y = 4 \][/tex]

Thus, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is [tex]\( 4 \)[/tex].
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