Question

which statements are true? check all that apply
line xy contains ray ox and ray oy
line xy contains segment yx
\\(overline{ca}\\) and \\(overline{ac}\\) are different segments
ray ba exisis in the diagram
ray yx exisis in the diagram

Explanation:

Step1: Analyze the first statement

A line is made up of rays. Line \(XY\) has points \(X\) and \(Y\). Ray \(OX\) and ray \(OY\) are part of line \(XY\) (since \(O\) is on line \(XY\)). But ray \(OX\) has endpoint \(O\) and goes through \(X\), ray \(OY\) has endpoint \(O\) and goes through \(Y\). However, line \(XY\) is a straight - line. A line is a set of points that extends infinitely in both directions. Ray \(OX\) and ray \(OY\) are part of line \(XY\). But the first statement is wrong because ray \(OX\) and ray \(OY\) have a common endpoint \(O\) on line \(XY\), but a line is not composed of two rays with a common interior point in the way described. A line can be thought of as a combination of two opposite - direction rays (e.g., ray \(XY\) and ray \(YX\)).

Step2: Analyze the second statement

A line segment \(YX\) is a part of the line \(XY\). A line segment has two endpoints. Since \(Y\) and \(X\) are points on line \(XY\), segment \(YX\) (with endpoints \(Y\) and \(X\)) is contained within line \(XY\).

Step3: Analyze the third statement

A line segment \(\overline{CA}\) has endpoints \(C\) and \(A\), and \(\overline{AC}\) has endpoints \(A\) and \(C\). But a line segment is determined by its two endpoints, and \(\overline{CA}\) and \(\overline{AC}\) are the same segment (the order of writing the endpoints of a line segment does not matter).

Step4: Analyze the fourth statement

For ray \(BA\), we need two points \(B\) and \(A\) such that the ray starts at \(B\) and passes through \(A\). Looking at the diagram, there is no such configuration where we can form ray \(BA\) (the points \(A\) and \(B\) are not in a position to form ray \(BA\) as described).

Step5: Analyze the fifth statement

Ray \(YX\) starts at \(Y\) and passes through \(X\). Since \(X\) and \(Y\) are points on the line (which is drawn), ray \(YX\) exists in the diagram.

Answer:

Line \(XY\) contains segment \(YX\); Ray \(YX\) exists in the diagram.