Question

partially correct
your answer is incorrect.

in the figure below, points ( y, s, r, ) and ( q ) lie in plane ( p ).
points ( t ) and ( x ) do not lie in plane ( p ).

for each part below, fill in the blanks to write a true statement.

your answer

(a) ( overleftrightarrow{tx} ) and ( overleftrightarrow{qs} ) are distinct lines that intersect.
(b) ( s, t, ) and ( q ) are distinct points that are collinear.
(c) another name for plane ( p ) is plane ( y ).
(d) point ( y ) and line ( overleftrightarrow{r} ) are coplanar.

Answer:

To solve this, we analyze each part using concepts of planes, lines, and collinearity:

Part (a)

• We need two distinct lines that intersect. Line \(\overleftrightarrow{TX}\) is the horizontal line through \(T\), \(R\), \(X\). Line \(\overleftrightarrow{QS}\) is the vertical line through \(Q\), \(R\), \(S\). They intersect at \(R\). So the correct line to pair with \(\overleftrightarrow{TX}\) is \(\overleftrightarrow{QS}\) (or \(\overleftrightarrow{QR}\), \(\overleftrightarrow{RS}\) etc., but \(\overleftrightarrow{QS}\) is a good choice as it includes \(Q\) and \(S\) in the plane). Wait, actually, looking at the figure, \(\overleftrightarrow{TX}\) and \(\overleftrightarrow{QR}\) (or \(\overleftrightarrow{QS}\)) intersect at \(R\). But the original answer had \(\overleftrightarrow{QS}\), maybe that's correct? Wait, no, let's re - check. The line \(\overleftrightarrow{TX}\) and \(\overleftrightarrow{QR}\) (since \(Q\), \(R\), \(S\) are colinear) intersect at \(R\). So \(\overleftrightarrow{TX}\) and \(\overleftrightarrow{QR}\) (or \(\overleftrightarrow{QS}\)) are distinct lines that intersect.

Part (b)

• Collinear points lie on the same line. Points \(S\), \(R\), and \(Q\) are on the vertical line (or \(S\), \(R\), \(T\)? Wait, no. \(T\) is not in plane \(P\), \(S\) and \(Q\) are in plane \(P\). Wait, the line \(\overleftrightarrow{QS}\) (or \(\overleftrightarrow{QR}\)) has \(Q\), \(R\), \(S\). But also, the line \(\overleftrightarrow{TX}\) has \(T\), \(R\), \(X\). Wait, the problem says "distinct points that are collinear". Let's see: \(S\), \(R\), and \(Q\) are collinear (on the vertical line in plane \(P\)), or \(S\), \(R\), \(T\)? No, \(T\) is not in plane \(P\), but the line \(\overleftrightarrow{TX}\) passes through \(R\) (which is in plane \(P\)) and \(S\) is in plane \(P\)? Wait, the figure shows that \(S\) is on the intersection of the vertical line (through \(Q\), \(R\), \(S\)) and the horizontal line (through \(T\), \(R\), \(X\))? Wait, no, the horizontal line is \(\overleftrightarrow{TX}\) (with \(T\) outside, \(R\) in the plane, \(X\) outside), and the vertical line is \(\overleftrightarrow{QS}\) (with \(Q\) in the plane, \(R\) in the plane, \(S\) in the plane). So the collinear points among \(S\), \(T\), \(Q\)? No, \(T\) is not on the line with \(S\) and \(Q\). Wait, the correct collinear points should be \(S\), \(R\), \(Q\) (on the vertical line) or \(S\), \(R\), \(T\) (on the horizontal line)? Wait, \(R\) is the intersection point of \(\overleftrightarrow{TX}\) and \(\overleftrightarrow{QS}\). So if we take \(S\), \(R\), and \(Q\), they are collinear. But the problem has \(S\), \(T\), and \(Q\) – that's incorrect. The correct point instead of \(T\) should be \(R\). So \(S\), \(R\), and \(Q\) are collinear.

Part (c)

• A plane can be named by three non - collinear points in the plane. Points \(Y\), \(S\), \(Q\) (or \(Y\), \(S\), \(R\), \(Q\)) are in plane \(P\). So another name for plane \(P\) can be plane \(YSQ\) (or plane \(YSR\), plane \(YQR\) etc.). The original answer had plane \(Y\) which is incorrect because a plane needs at least three non - collinear points (or a line and a point not on the line, or two intersecting lines). So we can use three points from \(Y\), \(S\), \(Q\), \(R\) to name the plane. For example, plane \(YSQ\).

Part (d)

• A point and a line are coplanar if the point lies in the plane that contains the line (or the line and the point lie in the same plane). Line \(\overleftrightarrow{TX}\) passes through \(R\) (in plane \(P\)) and \(T\), \(X\) (outside). But point \(Y\) is in plane \(P\), and line \(\overleftrightarrow{QS}\) (or \(\overleftrightarrow{QR}\), \(\overleftrightarrow{RS}\)) is in plane \(P\). Wait, the line in the blank: we need a line that is in plane \(P\) (since \(Y\) is in plane \(P\)). The line \(\overleftrightarrow{QS}\) (or \(\overleftrightarrow{QR}\), \(\overleftrightarrow{RS}\), \(\overleftrightarrow{YQ}\) – no, \(Y\) is a point, not on the vertical line? Wait, the figure shows \(Y\) is in plane \(P\), below \(S\). The line \(\overleftrightarrow{QS}\) is in plane \(P\), and \(Y\) is in plane \(P\), so \(Y\) and line \(\overleftrightarrow{QS}\) (or \(\overleftrightarrow{QR}\), \(\overleftrightarrow{RS}\), \(\overleftrightarrow{YQ}\) – no, better to use a line that \(Y\) and the line are in the same plane. Since \(Y\), \(S\), \(Q\), \(R\) are in plane \(P\), any line formed by two of these points (e.g., \(\overleftrightarrow{QS}\), \(\overleftrightarrow{SR}\), \(\overleftrightarrow{RQ}\), \(\overleftrightarrow{YS}\), \(\overleftrightarrow{YQ}\), \(\overleftrightarrow{YR}\)) will be coplanar with \(Y\). The original answer had line \(R\) which is incorrect (it's a point, not a line). We should use a line, e.g., \(\overleftrightarrow{QS}\) or \(\overleftrightarrow{SR}\) or \(\overleftrightarrow{RQ}\) or \(\overleftrightarrow{YS}\) or \(\overleftrightarrow{YQ}\) or \(\overleftrightarrow{YR}\) or \(\overleftrightarrow{TX}\) (no, \(TX\) passes through \(R\) in plane \(P\), but \(T\) and \(X\) are outside. Wait, \(Y\) is in plane \(P\), and line \(\overleftrightarrow{TX}\) passes through \(R\) (in plane \(P\)). So \(Y\) and line \(\overleftrightarrow{TX}\) – no, \(Y\) is in plane \(P\), and line \(\overleftrightarrow{TX}\) has a point \(R\) in plane \(P\), but \(T\) and \(X\) are outside. Wait, a line and a point are coplanar if the point lies on the line or the point and the line lie in a plane. Since \(Y\) is in plane \(P\) and line \(\overleftrightarrow{QS}\) is in plane \(P\), \(Y\) and line \(\overleftrightarrow{QS}\) are coplanar.

Correct Answers:

(a) \(\overleftrightarrow{TX}\) and \(\boldsymbol{\overleftrightarrow{QR}}\) (or \(\overleftrightarrow{QS}\), \(\overleftrightarrow{RS}\)) are distinct lines that intersect.
(b) \(S\), \(\boldsymbol{R}\), and \(Q\) are distinct points that are collinear.
(c) Another name for plane \(P\) is plane \(\boldsymbol{YSQ}\) (or plane \(YSR\), plane \(YQR\), etc. using three non - collinear points in plane \(P\)).
(d) Point \(Y\) and line \(\boldsymbol{\overleftrightarrow{QS}}\) (or \(\overleftrightarrow{QR}\), \(\overleftrightarrow{RS}\), \(\overleftrightarrow{YS}\), etc. – a line in plane \(P\)) are coplanar.

If we assume the intended answers (based on the figure's likely structure):

(a) \(\overleftrightarrow{TX}\) and \(\boldsymbol{\overleftrightarrow{QR}}\) (or \(\overleftrightarrow{QS}\)) intersect at \(R\).
(b) \(S\), \(\boldsymbol{R}\), \(Q\) are collinear (on the vertical line).
(c) Plane \(P\) can be named plane \(\boldsymbol{YSQ}\) (using points \(Y\), \(S\), \(Q\) in the plane).
(d) Point \(Y\) and line \(\boldsymbol{\overleftrightarrow{QS}}\) (in plane \(P\)) are coplanar.

For the purpose of filling the blanks as per the problem's structure (correcting the errors):

(a) \(\overleftrightarrow{TX}\) and \(\boldsymbol{\overleftrightarrow{QR}}\) (or \(\overleftrightarrow{QS}\)) are distinct lines that intersect.
(b) \(S\), \(\boldsymbol{R}\), and \(Q\) are distinct points that are collinear.
(c) Another name for plane \(P\) is plane \(\boldsymbol{YSQ}\) (or plane \(YSR\), etc.).
(d) Point \(Y\) and line \(\boldsymbol{\overleftrightarrow{QS}}\) (or \(\overleftrightarrow{QR}\), etc.) are coplanar.

If we have to give the most appropriate single - answer style for each blank (assuming the figure has \(\overleftrightarrow{TX}\) (horizontal) and \(\overleftrightarrow{QS}\) (vertical) intersecting at \(R\), \(Y\) in plane \(P\) with \(S\), \(Q\), \(R\)):

(a) \(\overleftrightarrow{TX}\) and \(\boldsymbol{\overleftrightarrow{QS}}\) are distinct lines that intersect.
(b) \(S\), \(\boldsymbol{R}\), and \(Q\) are distinct points that are collinear.
(c) Another name for plane \(P\) is plane \(\boldsymbol{YSQ}\).
(d) Point \(Y\) and line \(\boldsymbol{\overleftrightarrow{QS}}\) are coplanar.

Final Answers (Filled Blanks):

(a) \(\overleftrightarrow{TX}\) and \(\boldsymbol{\overleftrightarrow{QS}}\) are distinct lines that intersect.
(b) \(S\), \(\boldsymbol{R}\), and \(Q\) are distinct points that are collinear.
(c) Another name for plane \(P\) is plane \(\boldsymbol{YSQ}\).
(d) Point \(Y\) and line \(\boldsymbol{\overleftrightarrow{QS}}\) are coplanar.

(Note: The answers may vary slightly depending on the exact figure details, but the above are the corrections based on the given information and geometric principles.)