Question

A D (1 point) < < B C Lines AC and BD are diagonals of ABCD. What value of x makes ABCD a rectangle if ACx+7 and BD = 3x-5? Next▸

Answer 1

To determine the value of \( x \) that makes the quadrilateral \( ABCD \) a rectangle, we use the property that in a rectangle, the diagonals are equal in length.

Given:
- The length of diagonal \( AC \) is \( AC = x + 7 \)
- The length of diagonal \( BD \) is \( BD = 3x - 5 \)

Since the diagonals must be equal in a rectangle, we can set the expressions for the lengths equal to each other:

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

Now, we can solve for \( x \):

1. Subtract \( x \) from both sides:

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

This simplifies to:

\[
7 = 2x - 5
\]

2. Add \( 5 \) to both sides:

\[
7 + 5 = 2x
\]

\[
12 = 2x
\]

3. Divide by \( 2 \):

\[
x = \frac{12}{2} = 6
\]

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

\[
\boxed{6}
\]