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

Question

a. x = 15
b. x = 13
c. x = 6
d. x = 8

Answer 1

In a rectangle, the diagonals are equal in length. Therefore, we can set the lengths of the diagonals AC and BD equal to each other:

\[ AC = BD \]

Substituting the expressions for AC and BD, we get:

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

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

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

\[ 7 = 3x - x - 5 \] \[ 7 = 2x - 5 \]

2. Add 5 to both sides:

\[ 7 + 5 = 2x \] \[ 12 = 2x \]

3. Divide by 2:

\[ x = 6 \]

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

\[ \boxed{6} \]

Therefore, the correct answer is c. \(x = 6\).