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=15 x equals 15 x=8 x equals 8 x=13 x equals 13 x=6

Answer 1

In a rectangle, the diagonals are equal in length. Therefore, you can set the lengths of the diagonals equal to each other:

\[ AC = BD \]

Given that \( AC = x + 7 \) and \( BD = 3x - 5 \), we can set up the equation:

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

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

1. Subtract \( x \) from both sides:

\[ 7 = 2x - 5 \]

2. Add 5 to both sides:

\[ 12 = 2x \]

3. Divide both sides by 2:

\[ x = 6 \]

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

The correct answer is x = 6.