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=4
x equals 4

x=8
x equals 8

x=3

Answer 1

To determine the value of \( x \) that proves that parallelogram ABCD is a rectangle by showing that the diagonals are congruent, we need to set up the equation based on the information given.

Given:
- \( AC = 48 \)
- \( DO = 7x + 3 \)

In a parallelogram, the diagonals bisect each other, so \( AC \) and \( DO \) must be equal when the parallelogram is a rectangle.

Since \( O \) is the midpoint of both diagonals, we have:
\[
AC = 2 \times AO
\]
Since \( DO \) is also divided into two equal segments (DO = 2 * OD), we have:
\[
DO = 2 \times OD
\]

We can set \( DO \) equal to \( AC \) because they are equal when the parallelogram is a rectangle:
\[
48 = 7x + 3
\]

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

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

However, upon reconsidering the values provided as options, you will want to check against:
- \( 7x + 3 = 48 \)

This simplifies our earlier attempts, so using the options, we can test to see which value satisfies it:

1. For \( x = 6 \):
\[
7(6) + 3 = 42 + 3 = 45 \quad (\text{Incorrect})
\]

2. For \( x = 4 \):
\[
7(4) + 3 = 28 + 3 = 31 \quad (\text{Incorrect})
\]

3. For \( x = 8 \):
\[
7(8) + 3 = 56 + 3 = 59 \quad (\text{Incorrect})
\]

4. Lastly, for \( x = 3 \):
\[
7(3) + 3 = 21 + 3 = 24 \quad (\text{Incorrect})
\]

It appears there's a miscalculation. Therefore, if \( 7x = 48 - 3 \implies x = 6.43 \).

No provided options work as \( 6.43 \) doesn't match the solutions from provided options.

Thus, if we look closely through \( AC = 2AO \text{ and } DO = 2OD \), the answers provided likely intend to convey certain values without necessarily checking against the equality moved. Given your provided values, none suit as final confirmations to a parallelogram defined as a rectangle on those values.

**Conclusion: None of the given options are correct based on the values provided that \( DO = 7x + 3\) compared to the diagonals' relations.**