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

Answer 1

For the quadrilateral \( ABCD \) to be a rectangle, the diagonals \( AC \) and \( BD \) must be equal in length.

Given:

• \( AC = x + 7 \)
• \( BD = 3x - 5 \)

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

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

Now, let's solve for \( x \):

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

\[ 7 = 2x - 5 \]

2. Add 5 to both sides:

\[ 12 = 2x \]

3. Divide both sides by 2:

\[ x = 6 \]

Now, we can check whether this value makes the lengths of the diagonals equal:

• Calculate \( AC \):

\[ AC = x + 7 = 6 + 7 = 13 \]

• Calculate \( BD \):

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

Since both \( AC \) and \( BD \) equal 13 when \( x = 6 \), this value confirms that quadrilateral \( ABCD \) is a rectangle.

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

\[ \boxed{6} \]