Question

lines $\overline{ac}$ and $\overline{bd}$ are diagonals of $abcd$. what value of $x$ makes $abcd$ a rectangle if $ac = x + 7$ and $bd = 3x - 5$? (1 point) $\bigcirc x = 6$ $\bigcirc x = 8$ $\bigcirc x = 13$ $\bigcirc x = 15$

Explanation:

Step1: Use the property of rectangle diagonals

In a rectangle, the diagonals are equal. So, \(AC = BD\).

Step2: Set up the equation

Given \(AC=x + 7\) and \(BD = 3x-5\), we have the equation \(x + 7=3x - 5\).

Step3: Solve the equation

Subtract \(x\) from both sides: \(7=3x - 5-x\), which simplifies to \(7 = 2x-5\).
Add \(5\) to both sides: \(7 + 5=2x\), so \(12=2x\).
Divide both sides by \(2\): \(x=\frac{12}{2}=6\).

Answer:

\(x = 6\) (the first option)