Question

A parallelogram with vertices upper A upper B upper C upper D.
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

For the parallelogram ABCD to be a rectangle, the lengths of the diagonals AC and BD must be equal. Therefore, we can set the lengths equal to each other:

\[
AC = BD
\]

Given:
- \( AC = x + 7 \)
- \( BD = 3x - 5 \)

Setting the 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:

\[
x = 6
\]

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

\[
\boxed{6}
\]