Question

Use the image to answer the question.

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

Because congruent diagonals prove that a parallelogram is a rectangle, what value of x proves that parallelogram ABCD
is a rectangle if AC=48
and DO=7x+3
?

(1 point)
Responses

x=6
x equals 6

x=8
x equals 8

x=3
x equals 3

x=4

Answer 1

To determine the value of \( x \) that proves parallelogram ABCD is a rectangle, we need to use the information about the diagonals. Since the diagonals of a rectangle are congruent, we can set the lengths of the diagonals equal to each other.

Given:
- Length of diagonal AC = 48
- Length of diagonal DO (which is the same as diagonal AC, since they are opposite diagonals in a parallelogram) = \( 7x + 3 \)

Setting the lengths equal:
\[
7x + 3 = 48
\]

Now, solve for \( x \):
1. Subtract 3 from both sides:
\[
7x = 48 - 3
\]
\[
7x = 45
\]

2. Divide both sides by 7:
\[
x = \frac{45}{7}
\]
\[
x \approx 6.43
\]

Since none of the provided answers are \( \frac{45}{7} \), it seems there is an issue, as \( 7x + 3 \) must equal 48. Let's recheck the values in the options given, and cross-verify:

If we test each option:

1. If \( x = 6 \):
\[
7(6) + 3 = 42 + 3 = 45 \quad \text{(Not 48)}
\]

2. If \( x = 8 \):
\[
7(8) + 3 = 56 + 3 = 59 \quad \text{(Not 48)}
\]

3. If \( x = 3 \):
\[
7(3) + 3 = 21 + 3 = 24 \quad \text{(Not 48)}
\]

4. If \( x = 4 \):
\[
7(4) + 3 = 28 + 3 = 31 \quad \text{(Not 48)}
\]

None of the provided values yield 48 for the diagonal length \( DO \).

Thus the conclusion is that proper calculation should provide a correct equation setup to prove ABCD is a rectangle. However, something seems inconsistent as \( ax + b = c \) needs to yield an integer \( x \).

From what has been calculated, if no correct input matches, we would need to reconsider the problem or possibly there was a miscommunication for the values of lengths of the diagonals. Please verify if any updates or corrections can be made, thereafter the approach can be re-addressed. The response remains undetermined based on the congruent diagonals indicating the rectangle verification.

Can you confirm or check back on the data presented, for complete solution?