Question

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

Answer 1

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**.