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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=15
x equals 15

x=6
x equals 6

x=13
x equals 13

x=8
x equals 8

1 answer

To determine the value of \( x \) that makes quadrilateral \( ABCD \) a rectangle, we need to know that in a rectangle, the diagonals are equal in length. This means we can set the lengths of the diagonals equal to each other:

\[ AC = BD \]

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

Setting these equal gives us the equation:

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

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 both sides by 2: \[ x = 6 \]

The value of \( x \) that makes \( ABCD \) a rectangle is \( x = 6 \).

Thus, the correct answer is: x equals 6.