Question

in the figure below, points t, r, x, y, and z lie in plane p. points q and s do not lie in plane p. for each part below, fill in the blanks to write a true statement. (a) □ and x are distinct points that are collinear. (b) y, □, □, and □ are distinct points that are coplanar. (c) suppose line \overleftrightarrow{qt} is drawn on the figure. then \overleftrightarrow{qt} and □ are distinct lines that intersect. (d) another name for plane p is plane □.

Explanation:

(a)

Step1: Recall collinear points definition

Collinear points are points that lie on the same straight line. From the figure, \(Y\) (or \(Z\)) and \(X\) lie on the same line.

(b)

Step1: Recall coplanar points definition

Coplanar points are points that lie on the same plane. Points \(Y\), \(X\), \(Z\), \(T\) (or \(Y\), \(X\), \(Z\), \(R\)) lie on plane \(P\)

(c)

Step1: Analyze intersecting lines

If line \(\overleftrightarrow{QT}\) is drawn, line \(\overleftrightarrow{XR}\) (or \(\overleftrightarrow{XY}\), \(\overleftrightarrow{XZ}\)) intersects it.

(d)

Step1: Name a plane by three non - collinear points

A plane can be named by three non - collinear points on it. Plane \(P\) can be named as plane \(XRY\) (or other combinations of three non - collinear points from \(T\), \(R\), \(X\), \(Y\), \(Z\) that lie on plane \(P\))

Answer:

(a) \(Y\) (or \(Z\))
(b) \(X\), \(Z\), \(T\) (or \(X\), \(Z\), \(R\))
(c) \(\overleftrightarrow{XR}\) (or \(\overleftrightarrow{XY}\), \(\overleftrightarrow{XZ}\))
(d) \(XRY\) (or other valid combinations of three non - collinear points on plane \(P\))