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To solve this construction problem, let's go through the steps required to bisect angle \( \angle ABC \) and calculate the length of line segment \( CD \) accurately.

### Step-by-Step Construction Instructions:

1. **Draw \( \angle ABC \):**
   - \( \overline{AB} = 5 \text{ cm} \) and \( \overline{BC} = 5 \text{ cm} \).
   - The angle \( \angle ABC = 120^\circ \) is given.
   - Use a ruler to draw the lines, and a protractor to set the \( 120^\circ \) angle between the two lines at point \( B \).

2. **Constructing the Angle Bisector of \( \angle ABC \):**
   - Use your compass to construct the angle bisector:
     - Place the compass point at \( B \) and draw an arc that cuts both \( \overline{BA} \) and \( \overline{BC} \).
     - Label the intersections of the arc with \( \overline{BA} \) and \( \overline{BC} \) as points \( P \) and \( Q \), respectively.
     - Without adjusting the compass width, place the compass at points \( P \) and \( Q \) one at a time, and draw two arcs that intersect inside \( \angle ABC \). Label the intersection of these two arcs as \( R \).
     - Draw a straight line from \( B \) through \( R \). This is the bisector of \( \angle ABC \), and this line is \( \overline{BD} \).

3. **Measuring Length of \( CD \):**
   - Using the construction, we have \( BD = 4 \text{ cm} \) and \( CD = ? \). Since the drawing is not accurate and based on construction methods, we apply trigonometric reasoning to find \( CD \).
   
### Calculation of \( CD \):
In the triangle \( BCD \), we know:
- \( \overline{BC} = 5 \text{ cm} \),
- \( \overline{BD} = 4 \text{ cm} \),
- The angle at \( B \) is half of \( 120^\circ \), i.e., \( \angle CBD = 60^\circ \) (since \( BD \) bisects \( \angle ABC \)).

Using the **cosine rule** to find \( \overline{CD} \):
\[
CD^2 = BC^2 + BD^2 - 2 \times BC \times BD \times \cos(\angle CBD)
\]
Substitute the known values:
\[
CD^2 = 5^2 + 4^2 - 2 \times 5 \times 4 \times \cos(60^\circ)
\]
\[
CD^2 = 25 + 16 - 2 \times 5 \times 4 \times \left( \frac{1}{2} \right)
\]
\[
CD^2 = 25 + 16 - 20 = 21
\]
\[
CD = \sqrt{21} \approx 4.58 \, \text{cm}
\]

Thus, the length of \( CD \) is approximately **4.6 cm** when rounded to 1 decimal place.

### Do you need further details or clarifications? 
Here are some relative questions to further your understanding:
1. What is the importance of using the compass for accurate bisectors?
2. How would the length of \( CD \) change if \( BD \) was different?
3. Can you derive the cosine rule for solving general triangles?
4. How does bisecting the angle affect the angles in triangle \( BCD \)?
5. Why is rounding to 1 decimal place important in construction problems?

**Tip:** Always ensure your compass arcs are clear and unambiguous when constructing bisectors to avoid errors.

Use a ruler and a protractor to draw angle ABC as shown. Then, using only a ruler and a pair of compasses, construct line BD which bisects angle ABC. Measure the length of CD in your drawing to 1 d.p.

Learn how to bisect a 120° angle using a ruler and compass. This geometry problem includes steps for constructing an angle bisector and applying the cosine rule to calculate the length of CD. Suitable for students in Grades 8-10.

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