Question

in the figure below, points a, e, f, and c lie in plane x. points b and d do not lie in plane x. for each part below, fill in the blanks to write a true statement. (a) b, , and are distinct points that are collinear. (b) \\(\overrightarrow{fc}\\) and are distinct lines that intersect. (c) e, , , and are distinct points that are coplanar. (d) another name for plane x is plane .

Explanation:

Step1: Recall collinear points

Collinear points lie on the same line. Since there is a vertical line passing through \(B\) and \(D\), \(B\), \(D\) are collinear. We can choose any point on that line (but as there are no other points on it shown, we use \(D\)). So for (a), \(B\), \(D\) are collinear.

Step2: Find intersecting lines

Lines \(\overleftrightarrow{FC}\) and \(\overleftrightarrow{AC}\) (assuming there is a line passing through \(A\) and \(C\) which is a reasonable assumption as \(A\) and \(C\) are in the plane and can form a line) intersect at point \(C\). So for (b), \(\overleftrightarrow{FC}\) and \(\overleftrightarrow{AC}\) intersect.

Step3: Recall coplanar points

Coplanar points lie in the same plane. Points \(E\), \(F\), \(C\) and \(A\) all lie in plane \(X\). We can choose \(F\), \(C\) as other distinct points coplanar with \(E\). So for (c), \(E\), \(F\), \(C\) are coplanar.

Step4: Name the plane

A plane can be named by three non - collinear points in the plane. Points \(A\), \(C\), \(E\) are non - collinear points in plane \(X\). So for (d), another name for plane \(X\) is plane \(ACE\).

Answer:

(a) \(B\), \(D\)
(b) \(\overleftrightarrow{AC}\)
(c) \(F\), \(C\)
(d) \(ACE\)