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

Answer 1

For quadrilateral ABCD to be a rectangle, the lengths of the diagonals AC and BD must be equal. This means we can set the two expressions for the diagonals equal to each other:

\[ AC = BD \]

Given: \[ AC = x + 7 \] \[ BD = 3x - 5 \]

Setting the two expressions equal to each other:

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

Now, we can 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 \(x = 6\).

So the correct response is: x equals 6.