Question

1. if d is the midpoint of \\( \overline { c e } \\), \\( c d = x + 7 \\), and \\( c e = 5 x - 1 \\), determine each missing value. a. \\( x = \frac {? } {? } \\) b. \\( c d = \frac {? } {? } \\) c. \\( c e = \frac {? } {? } \\) d. \\( d e = \frac {? } {? } \\)

Explanation:

Step1: Use the mid - point property

Since \(D\) is the mid - point of \(\overline{CE}\), then \(CE = 2CD\).
Given \(CD=x + 7\) and \(CE = 5x-1\), we substitute into the equation \(CE = 2CD\):
\(5x-1=2(x + 7)\)

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(5x-1=2x+14\)
Subtract \(2x\) from both sides: \(5x-2x-1=2x-2x + 14\), so \(3x-1=14\)
Add \(1\) to both sides: \(3x-1 + 1=14+1\), then \(3x=15\)
Divide both sides by \(3\): \(x=\frac{15}{3}=5\)

Step3: Find the value of \(CD\)

Substitute \(x = 5\) into \(CD=x + 7\). Then \(CD=5 + 7=12\)

Step4: Find the value of \(CE\)

Substitute \(x = 5\) into \(CE = 5x-1\). Then \(CE=5\times5-1=25 - 1=24\)

Step5: Find the value of \(DE\)

Since \(D\) is the mid - point of \(\overline{CE}\), \(DE=CD\). So \(DE = 12\)

Answer:

a. \(x = 5\)
b. \(CD=12\)
c. \(CE = 24\)
d. \(DE=12\)