Question

use the figure and the given information to find the required measure.

\\( \overrightarrow { b d } \\) bisects \\( \angle a b c , a d = 3 x + 7 \\), and \\( c d = 2 x + 11 \\).

find \\( c d \\).

\\( c d = \square \\)

Explanation:

Step1: Use the Angle - Bisector Theorem (distance from a point to the sides of an angle)

If a point lies on the bisector of an angle, then it is equidistant from the sides of the angle. Here, since \(BD\) bisects \(\angle ABC\), \(AD = CD\).

Step2: Set up the equation

Given \(AD = 3x+7\) and \(CD = 2x + 11\), we set \(3x+7=2x + 11\).
Subtract \(2x\) from both sides: \(3x-2x+7=2x-2x + 11\), which gives \(x+7=11\).
Subtract \(7\) from both sides: \(x+7 - 7=11 - 7\), so \(x = 4\).

Step3: Find the value of \(CD\)

Substitute \(x = 4\) into the expression for \(CD\). Since \(CD=2x + 11\), then \(CD=2\times4+11\).
Calculate \(2\times4+11=8 + 11=19\).

Answer:

\(19\)