Question

point d is on line segment \\(\overline{ce}\\). given \\(de = x + 2\\), \\(cd = 3x + 2\\), and \\(ce = 5x + 2\\), determine the numerical length of \\(\overline{ce}\\).

Explanation:

Step1: Use segment addition postulate

Since D is on \( \overline{CE} \), we have \( CD + DE = CE \). Substituting the given expressions: \( (3x + 2) + (x + 2) = 5x + 2 \).

Step2: Simplify and solve for x

Simplify left side: \( 3x + 2 + x + 2 = 4x + 4 \). So equation is \( 4x + 4 = 5x + 2 \). Subtract \( 4x \) from both sides: \( 4 = x + 2 \). Subtract 2: \( x = 2 \).

Step3: Find length of \( \overline{CE} \)

Substitute \( x = 2 \) into \( CE = 5x + 2 \): \( 5(2) + 2 = 10 + 2 = 12 \).

Answer:

12