https://file.aiquickdraw.com/mathbot/file/17304711411109vwugfa4.jpg

To determine the correct answer, let's analyze each statement based on the coordinates of points \( A \), \( B \), \( C \), and \( D \) from the graph.

1. **Coordinates of Points**:
   - \( A = (-6, -3) \)
   - \( B = (-3, 0) \)
   - \( C = (0, 3) \)
   - \( D = (6, 6) \)

2. **Checking Slopes for \( \overline{AC} \) and \( \overline{BD} \)**:
   The slope formula is:
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   - For \( \overline{AC} \): Using points \( A = (-6, -3) \) and \( C = (0, 3) \),
     \[
     \text{slope of } \overline{AC} = \frac{3 - (-3)}{0 - (-6)} = \frac{6}{6} = 1
     \]
   - For \( \overline{BD} \): Using points \( B = (-3, 0) \) and \( D = (6, 6) \),
     \[
     \text{slope of } \overline{BD} = \frac{6 - 0}{6 - (-3)} = \frac{6}{9} = \frac{2}{3}
     \]

   The slopes of \( \overline{AC} \) and \( \overline{BD} \) are not the same, so **option 2 is false**.

3. **Checking Congruency of \( \overline{AC} \) and \( \overline{BD} \)**:
   To determine if the segments are congruent, calculate the lengths using the distance formula:
   \[
   \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   - For \( \overline{AC} \):
     \[
     \text{distance of } \overline{AC} = \sqrt{(0 - (-6))^2 + (3 - (-3))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
     \]
   - For \( \overline{BD} \):
     \[
     \text{distance of } \overline{BD} = \sqrt{(6 - (-3))^2 + (6 - 0)^2} = \sqrt{9^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117}
     \]

   Since \( \sqrt{72} \neq \sqrt{117} \), **\( \overline{AC} \) is not congruent to \( \overline{BD} \)**, making **option 3 false**.

4. **Checking if Points \( A \), \( B \), \( C \), and \( D \) Have the Same Coordinates**:
   Each point \( A \), \( B \), \( C \), and \( D \) has unique coordinates, so they do not share the same coordinates. Thus, **option 4 is false**.

5. **Conclusion**:
   Since all individual statements are false, the correct answer is:
   - **None of the choices are true** (since "All of the choices are true" is also not valid).

Would you like more detailed steps or clarifications? 

### Related Questions
1. What is the slope and how does it affect line orientation?
2. How can we determine congruency between two line segments?
3. What is the distance formula and how is it applied in geometry?
4. How does one identify unique coordinates on a coordinate plane?
5. How do line segments with different slopes affect parallelism?

### Tip
Remember, two line segments with different slopes will never be parallel or have the same steepness.

Based on the coordinates of points A, B, C, and D in the graph, determine if the following statements are true:
1. line segment AC and line segment BD have the same slope.
2. line segment AC is congruent to line segment BD.
3. A, B, C, and D have the same coordinates.
4. All of the choices are true.

Explore a coordinate geometry problem where we determine the slopes and lengths of line segments AC and BD using points A, B, C, and D. Learn to verify congruence and identify unique coordinates, ideal for students in Grades 8-10.

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