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) y, \square, \square, and \square are distinct points that are coplanar.
(b) suppose line \overleftrightarrow{qt} is drawn on the figure.
then \overleftrightarrow{qt} and \square are distinct lines that intersect.
(c) another name for plane p is plane \square.
(d) \square and x are distinct points that are collinear.

Explanation:

Part (a)

• Step1: Recall coplanar points definition

Coplanar points are points that lie on the same plane. Since \(T\), \(R\), \(X\), \(Y\), \(Z\) lie in plane \(P\). We can choose \(X\), \(Z\), \(T\) (answers may vary as long as the points are from \(T\), \(R\), \(X\), \(Y\), \(Z\)).

Part (b)

• Step1: Identify intersecting line

If we draw \(\overleftrightarrow{QT}\), and we know that \(T\) and \(R\) are related. \(\overleftrightarrow{TR}\) (or \(\overleftrightarrow{RT}\)) is a line in plane \(P\) that intersects \(\overleftrightarrow{QT}\) at point \(T\).

Part (c)

• Step1: Name the plane using non - collinear points

A plane can be named using three non - collinear points. Since \(X\), \(Y\), \(T\) (answers may vary as long as three non - collinear points from the plane \(P\) are used) lie in plane \(P\), we can use plane \(XYT\) (for example).

Part (d)

• Step1: Recall collinear points

Collinear points lie on the same line. Points \(Y\) and \(Z\) (or \(X\) and \(Z\)) lie on the same line as \(X\) (if we consider the line \(X - Y - Z\)).

Answer:

(a) \(X\), \(Z\), \(T\) (answers may vary)
(b) \(\overleftrightarrow{TR}\) (or \(\overleftrightarrow{RT}\))
(c) \(XYT\) (answers may vary)
(d) \(Z\) (or \(Y\))