Give two other names for WQ^(harr ). Give another name for plane V. Name three points that are collinear. Then name a fourth point that is not collinear with these three points. Name a point that is not coplanar with R,S and T.
The Correct Answer and Explanation is :
- Two other names for ( WQ^{\text{harr}} ):
- The symbol ( WQ^{\text{harr}} ) typically represents a line (because of the double-headed arrow), where W and Q are points on that line. So, other names for this line could be:
- ( \overleftrightarrow{WQ} ) (this denotes the line passing through points W and Q in both directions).
- ( \text{Line} WQ ) (another common way to describe a line passing through two points).
- Another name for plane ( V ):
- Planes are often named with a single capital letter (in this case, ( V )). An alternative name could be:
- ( \pi ) or any other letter, but we assume in geometric contexts that we stick to another unique identifier or a combination of three non-collinear points defining the plane. For example, if the plane ( V ) contains points ( R, S, ) and ( T ), we could name it ( \text{Plane} RST ).
- Three collinear points:
- If points are collinear, they lie on the same straight line. Let’s take the points ( W, Q, ) and ( P ). These three points are collinear because they all lie on the line ( WQ^{\text{harr}} ). So, we can say that the points ( W, Q, ) and ( P ) are collinear.
- A fourth point that is not collinear with these three points:
- For the points ( W, Q, ) and ( P ) to be collinear, any point not lying on the same line is non-collinear. Let’s take the point ( R ), which does not lie on the same line as ( W, Q, ) and ( P ). So, ( R ) is a point that is not collinear with the three points.
- A point that is not coplanar with ( R, S, ) and ( T ):
- Coplanar points are points that lie on the same plane. A point that is not coplanar with ( R, S, ) and ( T ) would be any point not lying on the same plane as these three points. Let’s name the point ( A ). If point ( A ) does not lie in the same plane as ( R, S, ) and ( T ), it is not coplanar with these points.
Now, let me illustrate these points and relationships visually. I’ll generate a diagram showing the geometric setup.
I wasn’t able to generate an image based on your request due to the content policy guidelines. However, I can still provide a detailed explanation of the geometric relationships and the names of the points and planes mentioned.
Here’s a breakdown of the geometric setup:
- Collinear Points: Points ( W ), ( Q ), and ( P ) are all on the same line ( WQ^{\text{harr}} ), meaning they are collinear. These three points lie in a straight line.
- Non-Collinear Point: Point ( R ) is not collinear with the points ( W ), ( Q ), and ( P ). It lies outside of the line defined by these three points.
- Coplanar Points: Points ( R ), ( S ), and ( T ) are on the same plane ( V ). Since they are coplanar, they lie within the same flat surface. You can imagine them as being on a flat sheet of paper.
- Non-Coplanar Point: Point ( A ) is not in the same plane as ( R ), ( S ), and ( T ). It is a point that lies outside of the plane ( V ), creating a 3D spatial separation.