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

Explanation:

Step1: Recall naming planes

A plane can be named by any three non - collinear points on the plane. Points $T$, $R$, $X$ are non - collinear points on plane $P$, so another name for plane $P$ is plane $TRX$.

Step2: Recall collinear points

Collinear points lie on the same line. Points $Y$ and $Z$ lie on the same line as $X$, so $Y$ (or $Z$) and $X$ are distinct points that are collinear.

Step3: Recall coplanar points

Coplanar points lie on the same plane. Points $Y$, $T$, $R$, $X$, $Z$ all lie on plane $P$. We can choose $T$, $R$, $X$ (or other non - collinear combinations) as distinct points that are coplanar with $Y$.

Step4: Recall intersecting lines

If we consider the line $\overleftrightarrow{QT}$, and assume we have another line that intersects it. Since $Q$ is outside the plane and $T$ is inside the plane, and we have lines in the plane, if we consider the line $\overleftrightarrow{RS}$ (as it is in the plane and can intersect $\overleftrightarrow{QT}$ at some point), $\overleftrightarrow{QT}$ and $\overleftrightarrow{RS}$ are distinct lines that intersect.

Answer:

(a) $TRX$
(b) $Y$ (or $Z$)
(c) $T$, $R$, $X$
(d) $\overleftrightarrow{RS}$