Question

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

Answer 1

To determine the value of \( x \) that makes quadrilateral \( ABCD \) a rectangle when given the lengths of the diagonals \( AC \) and \( BD \), we need to use the property that in a rectangle, the diagonals are equal in length.

We know that:

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

Setting these two expressions for the diagonals equal to each other, we have:

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

Next, we can solve for \( x \). First, we'll subtract \( x \) from both sides:

\[
7 = 2x - 5
\]

Then, add 5 to both sides:

\[
12 = 2x
\]

Now, divide both sides by 2:

\[
x = 6
\]

To ensure that this value is correct, we can substitute \( x = 6 \) back into the expressions for the lengths of the diagonals to check if they are equal:

Calculating \( AC \):

\[
AC = 6 + 7 = 13
\]

Calculating \( BD \):

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

Since both diagonals \( AC \) and \( BD \) are equal to 13, the diagonals confirmation is satisfied.

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

\[
\boxed{6}
\]