Question

Lines overline AC and overlineBD are diagonals of ABCD. What value of x makes ABCD a rectangle if AC = x+ 7 and
BD = 3x - 5?

Answer 1

For quadrilateral ABCD to be a rectangle, the lengths of the diagonals \( AC \) and \( BD \) must be equal. We have the expressions for the lengths of the diagonals:

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

To find the value of \( x \) that makes the diagonals equal, we set the two expressions equal to each other:

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

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

1. Subtract \( x \) from both sides:
\[
7 = 2x - 5
\]

2. Add 5 to both sides:
\[
12 = 2x
\]

3. Divide by 2:
\[
x = 6
\]

Now, we can box the solution:

\[
\boxed{6}
\]

To confirm that this value makes \( AC \) and \( BD \) equal, we can substitute \( x = 6 \) back into the expressions for \( AC \) and \( BD \):

\[
AC = 6 + 7 = 13
\]
\[
BD = 3(6) - 5 = 18 - 5 = 13
\]

Since \( AC = BD = 13 \), the solution is verified. Thus, the value of \( x \) that makes ABCD a rectangle is \( \boxed{6} \).