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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=8
x equals 8

x=13
x equals 13

x=15
x equals 15
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1 answer

To determine the value of \( x \) that makes the parallelogram \( ABCD \) a rectangle, we need to establish that the lengths of the diagonals \( AC \) and \( BD \) are equal, since in a rectangle, the diagonals are equal in length.

Given:

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

To find \( x \), we set these two expressions equal to each other:

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

Now, we will solve for \( x \):

  1. Subtract \( x \) from both sides: \[ 7 = 2x - 5 \]

  2. Add 5 to both sides: \[ 12 = 2x \]

  3. Divide by 2: \[ x = 6 \]

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

The correct response is:

  • x = 6