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

Answer 1

In a rectangle, the diagonals are equal in length. Therefore, we can set the two expressions for the diagonals equal to each other:

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

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

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

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

which simplifies to:

\[ 7 = 2x - 5 \]

2. Next, add 5 to both sides:

\[ 7 + 5 = 2x \]

which simplifies to:

\[ 12 = 2x \]

3. Now, divide both sides by 2:

\[ x = 6 \]

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

x = 6.