Educational worksheet on collinear and coplanar points with diagrams and questions for students to identify geometric relationships.

Let's solve the problem step by step based on the image you provided. The task is about determining whether given points are collinear or coplanar.

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🔹 Definitions:

- Collinear Points: Points that lie on the same straight line.
- Coplanar Points: Points that lie on the same plane (a flat surface).

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## ✔ Part A: Collinear Points

We need to determine if the given points lie on a single straight line.

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1) Points A, B, and C

```
A —— B —— C
```

All three points lie on the same straight line.

✔ Answer: Collinear

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2) Points P, Q, and R

```
R
/
/
P——Q
```

Points P and Q are on a line, but point R is above the line (not on the same straight line).

✘ Answer: Not collinear

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3) Points U and V

```
U •
\
\
V
```

There are only two points. Any two points are always collinear, because a line can be drawn through any two points.

✔ Answer: Collinear

> Note: Two points are always collinear.

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4) Points S and T

```
S —— T
```

They lie on a straight line.

✔ Answer: Collinear

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## ✔ Part B: Coplanar Points

We need to determine if the given points lie on the same plane.

All the diagrams show a parallelogram, which represents a plane. Any points shown within or on the parallelogram are assumed to lie in that plane unless otherwise indicated.

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1) Points B, C, and D

```
B
/\
/ \
C ———— D
```

All three points are inside or on the parallelogram → they lie in the same plane.

✔ Answer: Coplanar

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2) Points M, N, P, and Q

```
M • P •
\ /
\
N •
\
Q
```

Wait — looking at the diagram:
- M is outside the parallelogram
- P, N, Q are on or inside the parallelogram

But since M is outside the plane of the parallelogram, it does not lie on the same plane as the others.

✘ Answer: Not coplanar

> Unless specified, we assume the parallelogram defines a plane, and only points on or within it are coplanar.

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3) Points Q, R, S, T, and U

```
Q
/ \
/ \
S —— T —— R
\
\
U
```

All points are either inside or on the parallelogram → all lie in the same plane.

✔ Answer: Coplanar

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4) Points T, U, V, and W

```
T • W •
\ /
\
U •
\
V
```

All four points are inside or on the parallelogram → they lie in the same plane.

✔ Answer: Coplanar

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## ✔ Final Answers:

A) Collinear or Not?

1) Points A, B, and C are collinear
2) Points P, Q, and R are not collinear
3) Points U and V are collinear
4) Points S and T are collinear

B) Coplanar or Not?

1) Points B, C, and D are coplanar
2) Points M, N, P, and Q are not coplanar
3) Points Q, R, S, T, and U are coplanar
4) Points T, U, V, and W are coplanar

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