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A parallelogram with vertices upper A upper B upper C upper D.

Lines AC¯¯¯¯¯¯¯¯ and BD¯¯¯¯¯¯¯¯ are diagonals of ABCD. What value of x makes ABCD a rectangle if AC=x+7 and BD=3x−5?

(1 point)
Responses

x=6
x equals 6

x=15
x equals 15

x=8
x equals 8

x=13

1 answer

To determine the value of \( x \) that makes the parallelogram \( ABCD \) a rectangle, we need to know that the diagonals of a rectangle are equal in length. The given lengths of the diagonals are:

  • \( AC = x + 7 \)
  • \( BD = 3x - 5 \)

Setting these two expressions equal because the diagonals must be the same length in a rectangle, we have:

\[ x + 7 = 3x - 5 \]

Now, let's solve for \( x \).

  1. Start by subtracting \( x \) from both sides:

\[ 7 = 2x - 5 \]

  1. Next, add \( 5 \) to both sides:

\[ 12 = 2x \]

  1. Finally, divide both sides by \( 2 \):

\[ x = 6 \]

Thus, the value of \( x \) that makes the parallelogram \( ABCD \) a rectangle is \( x = 6 \).

The correct response is:

x=6 x equals 6