Question

1.1.d practice: measuring segments and angles

determine:
\\(\vec{db} \cap \overline{bc} =\\)
\\(\vec{ba} \cup \vec{bd} =\\)
\\(\overline{ad} \cap \vec{bd} =\\)
\\(\overline{ab} \cup \vec{bd} =\\)

Explanation:

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<reasoning>

Analyze the geometric elements on the line

We are given a line with four collinear points ordered from left to right: \(A\), \(B\), \(C\), and \(D\).

• \(\overrightarrow{DB}\) is the ray starting at \(D\) and extending infinitely to the left through \(B\) and \(A\).
• \(\overline{BC}\) is the line segment with endpoints \(B\) and \(C\).
• \(\overrightarrow{BA}\) is the ray starting at \(B\) and extending infinitely to the left through \(A\).
• \(\overrightarrow{BD}\) is the ray starting at \(B\) and extending infinitely to the right through \(C\) and \(D\).
• \(\overline{AD}\) is the line segment with endpoints \(A\) and \(D\).

Evaluate the first operation

Find the intersection:
\[
\overrightarrow{DB} \cap \overline{BC}
\]

• \(\overrightarrow{DB}\) covers all points to the left of \(D\), including \(C\), \(B\), and \(A\).
• \(\overline{BC}\) is the segment between \(B\) and \(C\).
• The overlapping region is the segment \(\overline{BC}\).

Evaluate the second operation

Find the union:
\[
\overrightarrow{BA} \cup \overrightarrow{BD}
\]

• \(\overrightarrow{BA}\) goes infinitely to the left starting from \(B\).
• \(\overrightarrow{BD}\) goes infinitely to the right starting from \(B\).
• Together, they cover the entire line through \(A\) and \(D\), which is represented by the line \(\overleftrightarrow{AD}\) (or \(\overleftrightarrow{AB}\)).

Evaluate the third operation

Find the intersection:
\[
\overline{AD} \cap \overrightarrow{BD}
\]

• \(\overline{AD}\) is the segment from \(A\) to \(D\).
• \(\overrightarrow{BD}\) starts at \(B\) and goes infinitely to the right.
• The overlapping region starts at \(B\) and ends at \(D\), which is the segment \(\overline{BD}\).

Evaluate the fourth operation

Find the union:
\[
\overline{AB} \cup \overrightarrow{BD}
\]

• \(\overline{AB}\) is the segment from \(A\) to \(B\).
• \(\overrightarrow{BD}\) starts at \(B\) and goes infinitely to the right.
• Combining them covers everything from \(A\) extending infinitely to the right, which is the ray \(\overrightarrow{AD}\) (or \(\overrightarrow{AB}\)).

</reasoning>

<answer>

Question 1

\[\overrightarrow{DB} \cap \overline{BC} = \overline{BC}\]

Question 2

\[\overrightarrow{BA} \cup \overrightarrow{BD} = \overleftrightarrow{AD}\]

Question 3

\[\overline{AD} \cap \overrightarrow{BD} = \overline{BD}\]

Question 4

\[\overline{AB} \cup \overrightarrow{BD} = \overrightarrow{AD}\]
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Analyze the geometric elements on the line

We are given a line with four collinear points ordered from left to right: \(A\), \(B\), \(C\), and \(D\).

• \(\overrightarrow{DB}\) is the ray starting at \(D\) and extending infinitely to the left through \(B\) and \(A\).
• \(\overline{BC}\) is the line segment with endpoints \(B\) and \(C\).
• \(\overrightarrow{BA}\) is the ray starting at \(B\) and extending infinitely to the left through \(A\).
• \(\overrightarrow{BD}\) is the ray starting at \(B\) and extending infinitely to the right through \(C\) and \(D\).
• \(\overline{AD}\) is the line segment with endpoints \(A\) and \(D\).

Evaluate the first operation

Find the intersection:
\[
\overrightarrow{DB} \cap \overline{BC}
\]

• \(\overrightarrow{DB}\) covers all points to the left of \(D\), including \(C\), \(B\), and \(A\).
• \(\overline{BC}\) is the segment between \(B\) and \(C\).
• The overlapping region is the segment \(\overline{BC}\).

Evaluate the second operation

Find the union:
\[
\overrightarrow{BA} \cup \overrightarrow{BD}
\]

• \(\overrightarrow{BA}\) goes infinitely to the left starting from \(B\).
• \(\overrightarrow{BD}\) goes infinitely to the right starting from \(B\).
• Together, they cover the entire line through \(A\) and \(D\), which is represented by the line \(\overleftrightarrow{AD}\) (or \(\overleftrightarrow{AB}\)).

Evaluate the third operation

Find the intersection:
\[
\overline{AD} \cap \overrightarrow{BD}
\]

• \(\overline{AD}\) is the segment from \(A\) to \(D\).
• \(\overrightarrow{BD}\) starts at \(B\) and goes infinitely to the right.
• The overlapping region starts at \(B\) and ends at \(D\), which is the segment \(\overline{BD}\).

Evaluate the fourth operation

Find the union:
\[
\overline{AB} \cup \overrightarrow{BD}
\]

• \(\overline{AB}\) is the segment from \(A\) to \(B\).
• \(\overrightarrow{BD}\) starts at \(B\) and goes infinitely to the right.
• Combining them covers everything from \(A\) extending infinitely to the right, which is the ray \(\overrightarrow{AD}\) (or \(\overrightarrow{AB}\)).

</reasoning>

<answer>

Question 1

\[\overrightarrow{DB} \cap \overline{BC} = \overline{BC}\]

Question 2

\[\overrightarrow{BA} \cup \overrightarrow{BD} = \overleftrightarrow{AD}\]

Question 3

\[\overline{AD} \cap \overrightarrow{BD} = \overline{BD}\]

Question 4

\[\overline{AB} \cup \overrightarrow{BD} = \overrightarrow{AD}\]
</answer>

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"question_type": "Multi-part",
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"Geometric Set Operations"
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