Question

standard: g.gsr.4 1-dok2
\\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c d } \\) intersect at point z.
which statement is true about the points?
points a, b, c, d, and z are all in the same plane because there are 3 points on the same line.
points a, b, c, d, and z are all not in the same plane because there are 3 points on the same line.
points a, b, c, d, and z are all in the same plane because any 3 points on the same line.
points a, b, c, d, and z are all not in the same plane because any 3 points not on the same line.

Explanation:

Step1: Recall the plane - point relationship

A plane can be determined by a line and a point not on the line, or by three non - collinear points. If two lines intersect (in this case, \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) intersect at \(Z\)), then all the points on these two lines (points \(A\), \(B\), \(C\), \(D\), and \(Z\)) lie on the same plane.

Step2: Analyze each option

• Option 1: The reason “because there are 3 points on the same line” is not the correct justification for all points being in the same plane. The key is the intersection of two lines (which gives a plane - two intersecting lines define a plane).
• Option 2: This is incorrect. Since two lines \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) intersect at \(Z\), all points \(A\), \(B\), \(C\), \(D\), \(Z\) are coplanar.
• Option 3: Any two intersecting lines (and thus all points on them) lie on a plane. When two lines \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) intersect at \(Z\), we can think of the plane determined by the two intersecting lines. Points \(A\), \(B\) are on one line, points \(C\), \(D\) are on the other line, and \(Z\) is the intersection point.
• Option 4: This is incorrect. As two lines \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CD}\) intersect (a non - parallel and non - skew relationship in 3 - D space, but in the context of basic geometry problems like this, we assume we are working in a plane - like or 3 - D space where two intersecting lines define a plane), all the points on the two lines are coplanar.

Answer:

Points \(A\), \(B\), \(C\), \(D\), and \(Z\) are all in the same plane because any two intersecting lines (and thus all points on them) define a plane. So the correct option is “Points \(A\), \(B\), \(C\), \(D\), and \(Z\) are all in the same plane because any 3 points on the same line” (assuming the intended meaning is related to two intersecting lines, and the option description has a minor wording issue. If we consider the plane - determination by two intersecting lines (which can be thought of in terms of three non - collinear points: for example, take a point from each line and the intersection point), the correct logical option is the third one).