Question

which line contains point x? select all that apply. a) line m b) $overleftrightarrow{xr}$ c) $overleftrightarrow{xy}$ d) $overleftrightarrow{yr}$

Explanation:

Step1: Analyze line m

Line \( m \) is a horizontal line passing through \( Y \) and \( R \). Point \( X \) lies on the same horizontal line (since \( X \), \( Y \), and \( R \) are colinear horizontally), so line \( m \) contains \( X \).

Step2: Analyze \( \overleftrightarrow{XR} \)

The line \( \overleftrightarrow{XR} \) is the horizontal line passing through \( X \), \( Y \), and \( R \), so it contains \( X \).

Step3: Analyze \( \overleftrightarrow{XY} \)

The line \( \overleftrightarrow{XY} \) is the horizontal segment extended, passing through \( X \) and \( Y \), so it contains \( X \).

Step4: Analyze \( \overleftrightarrow{YR} \)

The line \( \overleftrightarrow{YR} \) is the horizontal line passing through \( Y \) and \( R \), which also passes through \( X \) (since \( X \), \( Y \), \( R \) are colinear), so it contains \( X \)? Wait, no—wait, \( \overleftrightarrow{YR} \) is the same as line \( m \) and \( \overleftrightarrow{XR} \) and \( \overleftrightarrow{XY} \) in terms of the horizontal line. Wait, let's re - examine the diagram:

- Line \( m \): horizontal line through \( Y \), \( R \), and \( X \) (since \( X \) is on the same horizontal line as \( Y \) and \( R \)).
- \( \overleftrightarrow{XR} \): This is the line passing through \( X \) and \( R \), which is the same as line \( m \), so it contains \( X \).
- \( \overleftrightarrow{XY} \): This is the line passing through \( X \) and \( Y \), which is part of the horizontal line that contains \( X \), so it contains \( X \).
- \( \overleftrightarrow{YR} \): This is the line passing through \( Y \) and \( R \), which is the same horizontal line as \( X \), so it contains \( X \)? Wait, maybe my initial thought was wrong. Wait, the diagram: \( X \) is to the left of \( Y \), \( Y \) is on line \( m \), \( R \) is on line \( m \). So line \( m \) (horizontal) passes through \( X \), \( Y \), \( R \). \( \overleftrightarrow{XR} \) is the line through \( X \) and \( R \) (same as line \( m \)), \( \overleftrightarrow{XY} \) is the line through \( X \) and \( Y \) (same as line \( m \)), \( \overleftrightarrow{YR} \) is the line through \( Y \) and \( R \) (same as line \( m \)). Wait, but maybe the original check on \( C \) was correct, but also \( A \), \( B \), \( D \)? Wait, no, let's look again.

Wait, the problem is "which line contains point \( X \)". Let's define each:

- Line \( m \): It's a horizontal line. From the diagram, \( X \), \( Y \), \( R \) are on this horizontal line. So line \( m \) contains \( X \).
- \( \overleftrightarrow{XR} \): This is the line passing through \( X \) and \( R \), so it contains \( X \).
- \( \overleftrightarrow{XY} \): This is the line passing through \( X \) and \( Y \), so it contains \( X \).
- \( \overleftrightarrow{YR} \): This is the line passing through \( Y \) and \( R \), which is the same as the line passing through \( X \), \( Y \), \( R \), so it contains \( X \). Wait, but maybe the diagram has \( X \) on the horizontal line with \( Y \) and \( R \), so:

So the correct options are:

- A) line \( m \): Because line \( m \) is the horizontal line that includes \( X \), \( Y \), \( R \).
- B) \( \overleftrightarrow{XR} \): This line passes through \( X \) and \( R \), so it contains \( X \).
- C) \( \overleftrightarrow{XY} \): This line passes through \( X \) and \( Y \), so it contains \( X \).
- D) \( \overleftrightarrow{YR} \): This line passes through \( Y \) and \( R \), and since \( X \) is on the same line as \( Y \) and \( R \), it contains \( X \). Wait, but maybe I misread the diagram. Let's assume that the horizontal line is line \( m \), and \( X \), \( Y \), \( R \) are colinear on line \( m \). So:

Line \( m \) (A) contains \( X \) because \( X \) is on that horizontal line.

\( \overleftrightarrow{XR} \) (B) is the line through \( X \) and \( R \), so it contains \( X \).

\( \overleftrightarrow{XY} \) (C) is the line through \( X \) and \( Y \), so it contains \( X \).

\( \overleftrightarrow{YR} \) (D) is the line through \( Y \) and \( R \), and since \( X \) is on the same line as \( Y \) and \( R \), it contains \( X \). Wait, but maybe the original check on \( C \) was correct, but also others. Wait, perhaps the diagram shows that \( X \) is on the horizontal line with \( Y \) and \( R \), so all of A, B, C, D? No, maybe not. Wait, let's re - express:

A line is a straight path that extends infinitely in both directions.

- Line \( m \): Extends infinitely in both directions (horizontal), and \( X \) is on it. So A is correct.
- \( \overleftrightarrow{XR} \): Extends infinitely in both directions through \( X \) and \( R \), so it contains \( X \). So B is correct.
- \( \overleftrightarrow{XY} \): Extends infinitely in both directions through \( X \) and \( Y \), so it contains \( X \). So C is correct.
- \( \overleftrightarrow{YR} \): Extends infinitely in both directions through \( Y \) and \( R \), and since \( X \) is colinear with \( Y \) and \( R \), it contains \( X \). So D is correct? Wait, maybe the diagram has \( X \) on the horizontal line, so all of A, B, C, D. But maybe I made a mistake. Wait, the original check on \( C \) was correct, but let's see:

Wait, the problem says "select all that apply". Let's analyze each option:

- Option A: Line \( m \). From the diagram, line \( m \) is the horizontal line with \( Y \) and \( R \), and \( X \) is on the same horizontal line (to the left of \( Y \)), so line \( m \) contains \( X \). So A is correct.
- Option B: \( \overleftrightarrow{XR} \). This is the line passing through \( X \) and \( R \), so it contains \( X \). So B is correct.
- Option C: \( \overleftrightarrow{XY} \). This is the line passing through \( X \) and \( Y \), so it contains \( X \). So C is correct.
- Option D: \( \overleftrightarrow{YR} \). This is the line passing through \( Y \) and \( R \), and since \( X \), \( Y \), \( R \) are colinear, this line also contains \( X \). So D is correct.

Wait, but maybe the diagram is such that \( \overleftrightarrow{YR} \) is the same as line \( m \), \( \overleftrightarrow{XR} \) is the same as line \( m \), \( \overleftrightarrow{XY} \) is part of line \( m \). So all A, B, C, D? But maybe the original check on \( C \) was a start, but let's confirm.

Answer:

A) line \( m \), B) \( \overleftrightarrow{XR} \), C) \( \overleftrightarrow{XY} \), D) \( \overleftrightarrow{YR} \) (assuming \( X \), \( Y \), \( R \) are colinear on the horizontal line. If there is a misinterpretation, but based on the diagram description, these lines contain \( X \)).