Question

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) \\( \bigcirc \\ x = 3 \\) \\( \bigcirc \\ x = 8 \\) \\( \bigcirc \\ x = 4 \\) \\( \bigcirc \\ x = 6 \\)

Explanation:

Step1: Recall rectangle diagonal property

In a rectangle (which is a parallelogram), diagonals are congruent and bisect each other. So, the diagonal \( AC \) is bisected by the other diagonal \( BD \), meaning \( AC = 2 \times DO \).

Step2: Substitute known values

We know \( AC = 48 \) and \( DO = 7x + 3 \). Substitute into the equation \( AC = 2 \times DO \):
\( 48 = 2(7x + 3) \)

Step3: Solve for x

First, divide both sides by 2:
\( \frac{48}{2}=7x + 3 \)
\( 24 = 7x + 3 \)
Then, subtract 3 from both sides:
\( 24 - 3 = 7x \)
\( 21 = 7x \)
Finally, divide both sides by 7:
\( x=\frac{21}{7}=3 \)

Wait, there is a mistake above. Wait, in a parallelogram (including rectangle), diagonals bisect each other. So \( AC = BD \), and \( BD = 2 \times DO \) (since diagonals bisect each other, so \( DO=\frac{BD}{2}\), so \( BD = 2DO \)). But also, in a rectangle, diagonals are equal, so \( AC = BD \). So \( AC = 2DO \). Wait, but let's check again.

Wait, \( AC = 48 \), so \( BD = 48 \) (since diagonals of rectangle are equal). Then, since diagonals bisect each other, \( DO=\frac{BD}{2}=\frac{48}{2}=24 \). So set \( 7x + 3 = 24 \). Then \( 7x = 24 - 3 = 21 \), so \( x = 3 \). Wait, but let's check the options. Wait, the options include \( x = 3 \), \( x = 8 \), \( x = 4 \), \( x = 6 \). Wait, but maybe I made a mistake. Wait, maybe the diagonal is \( BD \), and \( AC = BD \), and \( DO \) is half of \( BD \). So \( BD = 2 \times DO \), and \( AC = BD \), so \( AC = 2 \times DO \). So \( 48 = 2(7x + 3) \). Then \( 24 = 7x + 3 \), \( 7x = 21 \), \( x = 3 \). So the answer should be \( x = 3 \).

Wait, but let's re-express:

In a parallelogram, diagonals bisect each other. So, if \( ABCD \) is a parallelogram, then the diagonals \( AC \) and \( BD \) intersect at point \( O \), so \( AO = OC \) and \( BO = OD \). In a rectangle, diagonals are equal, so \( AC = BD \). Therefore, \( BD = AC = 48 \), so \( OD=\frac{BD}{2}=\frac{48}{2}=24 \). So we set \( 7x + 3 = 24 \). Solving for \( x \):

\( 7x + 3 = 24 \)

Subtract 3 from both sides:

\( 7x = 24 - 3 = 21 \)

Divide both sides by 7:

\( x = \frac{21}{7} = 3 \)

So the value of \( x \) is 3.

Answer:

\( x = 3 \) (corresponding to the option " \( x = 3 \)")