use the diagram. (See Example 2.) t B S A E C D 11. What is another name for \overline{BD}? 12. What is another name for \overline{AC}? 13. What is another name for ray \overrightarrow{AE}? 14. Name all rays with endpoint E. 15. Name two pairs of opposite rays
The Correct Answer and Explanation is:
Based on the provided diagram and the labeled points A,B,C,D,EA, B, C, D, E, and the lines and rays shown, here are the correct answers to Exercises 11–15, followed by a full explanation.
11. What is another name for BD‾\overline{BD}?
Answer: DB‾\overline{DB}
12. What is another name for AC‾\overline{AC}?
Answer: CA‾\overline{CA}
13. What is another name for ray AE→\overrightarrow{AE}?
Answer: AS→\overrightarrow{AS}
14. Name all rays with endpoint E.
Answer:
- EB→\overrightarrow{EB}
- ED→\overrightarrow{ED}
- EA→\overrightarrow{EA}
- EC→\overrightarrow{EC}
15. Name two pairs of opposite rays.
Answer:
- EA→\overrightarrow{EA} and EC→\overrightarrow{EC}
- ED→\overrightarrow{ED} and EB→\overrightarrow{EB}
Explanation:
In geometry, understanding lines, line segments, rays, and their relationships is essential for interpreting diagrams accurately.
For question 11, BD‾\overline{BD} is a line segment connecting points B and D. Since line segments are not directional, the order of points does not matter. Thus, BD‾=DB‾\overline{BD} = \overline{DB}. Both represent the same segment.
In question 12, the same logic applies. Segment AC‾\overline{AC} can also be written as CA‾\overline{CA}, since the distance and endpoints are unchanged.
Question 13 involves naming a ray. A ray starts at one point and extends infinitely in one direction. Ray AE→\overrightarrow{AE} begins at point A and passes through point E toward the right. Looking at the diagram, the ray AS→\overrightarrow{AS} also starts at A and moves in the same direction, since S lies on the same path before A, making it another name for the same ray.
For question 14, all rays with endpoint E include those extending from E through another point. From the diagram, E connects directly to B, D, A, and C, giving us rays EB→\overrightarrow{EB}, ED→\overrightarrow{ED}, EA→\overrightarrow{EA}, and EC→\overrightarrow{EC}.
Finally, question 15 asks about opposite rays—two rays that share the same endpoint and extend in exactly opposite directions forming a straight line. From the diagram:
- EA→\overrightarrow{EA} and EC→\overrightarrow{EC} are opposite rays on the horizontal line.
- ED→\overrightarrow{ED} and EB→\overrightarrow{EB} are opposite rays on the diagonal line.
Understanding these relationships helps develop spatial reasoning and accurate geometric labeling skills.
