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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=13 x equals 13 x=8 x equals 8 x=15

1 answer

To determine the value of \( x \) that makes quadrilateral \( ABCD \) a rectangle, we need to use the property that the diagonals of a rectangle are equal in length.

Given:

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

Setting the two lengths equal, we have:

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

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

  1. First, subtract \( x \) from both sides:

\[ 7 = 2x - 5 \]

  1. Next, add 5 to both sides:

\[ 12 = 2x \]

  1. Now, divide both sides by 2:

\[ x = 6 \]

Thus, the value of \( x \) that makes \( ABCD \) a rectangle is:

\[ \boxed{6} \]