Question

Use the image to answer the 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
x equals 15
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Answer 1

To determine the value of \( x \) that makes the parallelogram \( ABCD \) a rectangle, we need to establish that the lengths of the diagonals \( AC \) and \( BD \) are equal, since in a rectangle, the diagonals are equal in length.

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

To find \( x \), we set these two expressions equal to each other:

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

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

The correct response is:
- **x = 6**