Question

1. if b is the midpoint of \\( \overline { a c } \\), \\( a c = c d \\), \\( a b = 3 x + 4 \\), \\( a c = 11 x - 17 \\), and \\( c e = 49 \\), find de.

Explanation:

Step1: Use the mid - point property

Since \(B\) is the mid - point of \(\overline{AC}\), \(AC = 2AB\).
Given \(AB=3x + 4\) and \(AC = 11x-17\), we have the equation \(11x-17=2(3x + 4)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(11x-17 = 6x+8\).
Subtract \(6x\) from both sides: \(11x-6x-17=6x - 6x+8\), so \(5x-17 = 8\).
Add \(17\) to both sides: \(5x-17 + 17=8 + 17\), then \(5x=25\).
Divide both sides by \(5\): \(x=\frac{25}{5}=5\).

Step3: Find the length of \(AC\)

Substitute \(x = 5\) into the formula for \(AC\): \(AC=11x-17\).
\(AC=11\times5-17=55 - 17=38\).

Step4: Use the given \(AC = CD\)

Since \(AC = CD\), then \(CD = 38\).

Step5: Find the length of \(DE\)

We know that \(CE=CD + DE\).
Given \(CE = 49\) and \(CD = 38\), then \(DE=CE - CD\).
\(DE=49-38 = 11\).

Answer:

\(11\)