Question

use the image to answer the question.
lines \\( \overline { a c } \\) and \\( \overline { b d } \\) are diagonals of \\( a b c d \\). what value of \\( x \\) makes \\( a b c d \\) a rectangle if \\( a c = x + 7 \\) and \\( b d = 3 x - 5? \\)
(1 point)
\\( x = 15 \\)
\\( x = 6 \\)
\\( x = 8 \\)
\\( x = 13 \\)

Explanation:

Step1: Use the property of rectangle diagonals

In a rectangle, the diagonals are equal. So, \(AC = BD\).
Given \(AC=x + 7\) and \(BD = 3x-5\), we set up the equation \(x + 7=3x-5\).

Step2: Solve the equation for \(x\)

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

Answer:

\(x = 6\)