SOLUTION: Points Lines and Planes Worksheet - Studypool - Free Printable

Let's go through each question in the worksheet and verify or correct the answers based on the diagrams and geometric principles. I'll explain each answer clearly.

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Problem 1:

Diagram: Three lines intersecting:
- Line $ p $: passes through points $ M, N, O, P $
- Line $ q $: passes through $ J, K, L $
- Line $ r $: passes through $ H, N, K, I $

#### a) Four collinear points: H, N, K, I

✔ Correct.
Points $ H, N, K, I $ all lie on line $ r $. So they are collinear.

#### b) A line that contains point $ M $: line $ p $

✔ Correct.
Point $ M $ lies on line $ p $.

#### c) A line that contains points $ H $ and $ K $: line $ r $

✔ Correct.
Both $ H $ and $ K $ lie on line $ r $.

#### d) Another name for line $ q $: $ \overleftrightarrow{JL} $, $ \overleftrightarrow{JK} $, $ \overleftrightarrow{KL} $

✔ Correct.
Line $ q $ goes through $ J, K, L $. Any two of these points can name the line.

#### e) The intersection of lines $ p $ and $ r $: N

✔ Correct.
Lines $ p $ and $ r $ intersect at point $ N $.

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Problem 2:

Diagram: A plane $ M $ with points $ A, B, C, D, E, F $, and lines $ j $, $ k $, and $ l $.
- Line $ j $: passes through $ D, E, F $
- Line $ k $: vertical line through $ B, E, C $
- Plane $ M $: the rectangle-like surface

#### a) A line containing point $ F $: line $ j $

✔ Correct.
Point $ F $ is on line $ j $.

#### b) Another name for line $ k $: $ \overleftrightarrow{BE}, \overleftrightarrow{BC}, \overleftrightarrow{EC} $

✔ Correct.
Line $ k $ goes through $ B, E, C $. So any two of these points can name it.

#### c) A plane containing point $ A $: plane $ M $

✔ Correct.
Point $ A $ lies within plane $ M $.

#### d) An example of three non-collinear points: A, B, D

✔ Correct.
Points $ A, B, D $ do not lie on the same line — they form a triangle in the plane, so they are non-collinear.

#### e) The intersection of plane $ M $ and line $ k $: E

✔ Correct.
Line $ k $ passes through point $ E $, which lies in plane $ M $. Since line $ k $ only intersects plane $ M $ at $ E $ (as $ B $ and $ C $ are outside), the intersection is point $ E $.

> Note: If $ B $ and $ C $ are *on* the plane, then the entire segment $ BC $ lies in the plane, so the intersection would be the whole line $ k $. But since $ B $ and $ C $ are shown as extending vertically beyond the plane, likely only $ E $ is in the plane → so E is correct.

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Problem 3:

Diagram: A pyramid-like figure with:
- Point $ X $ at the top
- Base triangle $ YZT $
- Planes: $ XYZ $, $ XYP $, $ XTP $, $ YZT $, etc.
- Points: $ X, Y, Z, T, P, Q, R $, etc.

Assuming:
- Plane $ R $: the base plane $ YZT $
- Plane $ XYZ $: face from $ X $ to $ Y $ and $ Z $
- Plane $ XYP $: another face
- Plane $ XTP $: another face
- Plane $ YPT $: possibly side face?

But based on standard labeling:

#### a) Three coplanar points: Y, Z, T

✔ Correct.
These three points form the base triangle, so they are coplanar in plane $ R $.

#### b) A plane containing point $ X $: plane $ R $?

✘ Incorrect.
Wait — point $ X $ is not on plane $ R $ unless $ R $ is the top face. But typically, if $ R $ is labeled as the base, then $ X $ is above it.

So this answer is wrong.

Correction:
Plane $ XYZ $, $ XYP $, or $ XTP $ — any face containing $ X $.

✔ Correct answer: plane $ XYZ $ or plane $ XYP $ or plane $ XTP $.

The answer given "plane $ R $" is incorrect unless $ R $ is defined as a plane containing $ X $, which seems unlikely.

So: ✘ Incorrect.

#### c) The intersection of plane $ R $ and plane $ ZYT $: $ \overline{YZ} $

Wait — plane $ ZYT $? That’s the same as plane $ YZT $, which is the base.

So both planes are the same? Or is there confusion?

Wait — perhaps “plane $ R $” is the base (triangle $ YZT $), and “plane $ ZYT $” is just another way to write the same thing.

But if both are the same plane, their intersection is the whole plane.

But the answer says $ \overline{YZ} $, which is a line.

This suggests that plane $ R $ and plane $ ZYT $ are different.

But if $ ZYT $ is the same as $ YZT $, and $ R $ is the base, then they’re the same.

Possibility: Maybe plane $ R $ is a different plane?

Alternatively, maybe plane $ R $ is the front face (e.g., $ XYP $), and plane $ ZYT $ is the base.

Then their intersection would be line $ YT $?

But the answer says $ \overline{YZ} $ — which connects $ Y $ and $ Z $.

Let’s suppose:
- Plane $ R $: the base plane $ YZT $
- Plane $ ZYT $: same as $ YZT $

Then their intersection is the entire plane — but that’s not a line.

So unless the question means something else...

Wait — perhaps the diagram shows:
- Plane $ R $: one of the triangular faces (e.g., $ XYP $)
- Plane $ ZYT $: the base

Then their intersection is line $ YT $?

But the answer says $ \overline{YZ} $

Hmm.

Wait — maybe plane $ R $ is the plane $ XYP $, and plane $ ZYT $ is the base $ YZT $, and they intersect along line $ YT $? But $ YT $ is not named.

But the answer says $ \overline{YZ} $ — which is part of the base.

Only if both planes contain $ Y $ and $ Z $, and no other common points.

But if plane $ R $ is $ XYP $, it doesn’t contain $ Z $.

So unless $ R $ is the base, and $ ZYT $ is another name for the base, then they are the same plane.

So their intersection is the whole plane, not a line.

Therefore, the answer $ \overline{YZ} $ is incorrect.

But let’s look again: the student wrote $ \overline{YZ} $.

Possibility: Maybe plane $ R $ is the front face $ XYP $, and plane $ ZYT $ is the back face $ XZT $? No.

Alternatively, perhaps the question is asking for the intersection of plane $ R $ and plane $ XYT $?

But it says plane $ ZYT $.

Maybe typo?

Alternatively, perhaps plane $ R $ is the base, and plane $ ZYT $ is a different plane?

I think the intended meaning is:
- Plane $ R $: the base $ YZT $
- Plane $ ZYT $: same as $ YZT $

So intersection is the whole plane → not a line.

But the answer is $ \overline{YZ} $ — so likely incorrect.

But wait — maybe plane $ R $ is not the base.

Looking at the diagram: point $ X $ is above, and there are several triangles.

Perhaps:
- Plane $ R $: triangle $ XYP $
- Plane $ ZYT $: triangle $ YZT $

Then their intersection is point $ Y $?

But answer is $ \overline{YZ} $ — not matching.

Alternatively, maybe the student meant plane $ XYZ $ and plane $ YZT $?

Then their intersection is line $ YZ $.

Yes! That makes sense.

So likely, the question is:
> The intersection of plane $ XYZ $ and plane $ YZT $: $ \overline{YZ} $

But the question says: plane $ R $ and plane $ ZYT $

If plane $ R $ is plane $ XYZ $, and plane $ ZYT $ is plane $ YZT $, then yes, they intersect along $ \overline{YZ} $.

So if plane $ R $ = plane $ XYZ $, then ✔ correct.

But earlier, we thought plane $ R $ was the base — but maybe it's labeled differently.

Given the ambiguity, and since $ \overline{YZ} $ is a valid intersection of two planes sharing points $ Y $ and $ Z $, and assuming plane $ R $ is $ XYZ $, then answer is correct.

So: ✔ Acceptable, assuming plane $ R $ is $ XYZ $.

#### d) How many planes appear in the figure? 5

Let’s count:
1. Base: $ YZT $
2. Face $ XYZ $
3. Face $ XYP $
4. Face $ XTP $
5. Face $ YPT $? Or $ XZT $?

Wait — points: $ X, Y, Z, T, P $

From diagram:
- Triangle $ XYZ $
- Triangle $ XYP $
- Triangle $ XTP $
- Triangle $ YZT $
- Triangle $ YPT $? Or $ ZTP $?

Possibly:
- Plane $ XYZ $
- Plane $ XYP $
- Plane $ XTP $
- Plane $ YZT $
- Plane $ YPT $ or $ ZPT $

That’s 5 planes.

✔ Correct.

#### e) How many planes contain point $ W $? 3

But wait — point $ W $ is not labeled in the diagram!

Wait — in the diagram, is there a point $ W $? Let me check.

In the image: points labeled are $ X, Y, Z, T, P, Q, R $? Wait — the diagram has:
- $ X $ at top
- $ Y, Z, T $ on base
- $ P $ somewhere?
- $ Q, R $? Possibly labels on edges?

But the question asks about point $ W $ — but no such point is visible.

Possibility: Typo? Maybe it's point $ P $?

Or maybe $ W $ is a point on an edge?

Wait — in the diagram, there’s a point labeled $ W $ near the center of the base?

No — looking carefully, the labels are:
- $ X $ at top
- $ Y, Z, T $ forming base
- $ P $ on edge $ XT $
- $ Q $ on edge $ YT $
- $ R $ on edge $ ZT $
- $ W $? Not visible.

But the student wrote: 3

Possibility: Point $ P $ lies on multiple planes.

For example:
- Plane $ XYP $
- Plane $ XTP $
- Plane $ YPT $?

So 3 planes.

But the question says point $ W $ — likely a typo.

If it’s point $ P $, then yes — it lies on:
1. Plane $ XYP $
2. Plane $ XTP $
3. Plane $ YPT $ (if that exists)

So 3 planes.

But since $ W $ is not labeled, this is likely a mistake.

But assuming the answer is 3, and it refers to a point like $ P $, then ✔ plausible.

But strictly speaking, no point $ W $ is labeled — so error in question or diagram.

So: ⚠️ Unclear — but if $ W $ is a vertex or intersection point, and it lies in 3 planes, then 3 might be correct.

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Problem 4:

Diagram: Two planes $ P $ and $ Q $ intersecting along line $ l $, and line $ m $ passing through them.

- Line $ l $: horizontal line where planes $ P $ and $ Q $ intersect
- Line $ m $: diagonal line in plane $ Q $, passing through point $ E $
- Point $ D $ on plane $ P $
- Point $ E $ on plane $ Q $
- Point $ G $ on plane $ Q $
- Plane $ Q $ also labeled as plane $ TFG $?

#### a) The intersection of lines $ l $ and $ m $: E

✔ Correct.
Lines $ l $ and $ m $ intersect at point $ E $.

#### b) Another name for plane $ Q $: plane $ TFG $

✔ Correct.
Plane $ Q $ contains points $ T, F, G $, so it can be named plane $ TFG $.

#### c) Are points $ D $ and $ E $ collinear or coplanar?

Answer: collinear

✘ Incorrect.

- Points $ D $ and $ E $ are not on the same line — they are on different planes.
- But they are coplanar — because both lie in the space, and there exists a plane containing both (e.g., plane $ P $ and $ Q $ intersect — but $ D $ is in $ P $, $ E $ in $ Q $, and both on line $ l $? Wait.)

Wait — does point $ D $ lie on line $ l $? Yes — since $ D $ is on plane $ P $, and line $ l $ is the intersection of $ P $ and $ Q $, and $ D $ is on $ l $? From diagram: $ D $ is on plane $ P $, and appears to be on line $ l $.

Similarly, $ E $ is on line $ m $, which intersects $ l $ at $ E $.

So if $ D $ is on $ l $, and $ E $ is on $ l $, then both $ D $ and $ E $ are on line $ l $ → they are collinear.

So yes, they are collinear.

But are they necessarily collinear?

Let’s see: if $ D $ is on line $ l $, and $ E $ is on line $ l $, then yes — they are on the same line.

So "collinear" is correct.

✔ Correct.

#### d) How many lines do planes $ P $ and $ Q $ intersect? 0

✘ Incorrect.

Planes $ P $ and $ Q $ intersect along line $ l $ — so they intersect in one line.

So the answer should be 1, not 0.

✔ Correct answer: 1

But student wrote 0 — so ✘ Wrong.

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✔ Final Summary of Corrections:

| Problem | Question | Student Answer | Correct? | Correction |
|--------|---------|----------------|----------|------------|
| 1a | Four collinear points | H, N, K, I | ✔ | Correct |
| 1b | Line with M | line p | ✔ | Correct |
| 1c | Line with H and K | line r | ✔ | Correct |
| 1d | Another name for line q | JL, JK, KL | ✔ | Correct |
| 1e | Intersection of p and r | N | ✔ | Correct |
| 2a | Line with F | line j | ✔ | Correct |
| 2b | Another name for line k | BE, BC, EC | ✔ | Correct |
| 2c | Plane with A | plane M | ✔ | Correct |
| 2d | Non-collinear points | A, B, D | ✔ | Correct |
| 2e | Intersection of plane M and line k | E | ✔ | Correct |
| 3a | Coplanar points | Y, Z, T | ✔ | Correct |
| 3b | Plane with X | plane R | ✘ | Should be plane XYZ or similar |
| 3c | Intersection of plane R and plane ZYT | YZ | ⚠️ | Only correct if plane R is XYZ and plane ZYT is YZT |
| 3d | Number of planes | 5 | ✔ | Likely correct |
| 3e | Planes through W | 3 | ⚠️ | Point W not labeled; likely typo (should be P?) |
| 4a | Intersection of l and m | E | ✔ | Correct |
| 4b | Another name for plane Q | plane TFG | ✔ | Correct |
| 4c | D and E: collinear or coplanar? | collinear | ✔ | Correct (since both on line l) |
| 4d | How many lines do P and Q intersect? | 0 | ✘ | Should be 1 |

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✔ Final Answers (Corrected):

#### Problem 1:
a) H, N, K, I
b) line p
c) line r
d) $ \overleftrightarrow{JL} $, $ \overleftrightarrow{JK} $, $ \overleftrightarrow{KL} $
e) N

#### Problem 2:
a) line j
b) $ \overleftrightarrow{BE} $, $ \overleftrightarrow{BC} $, $ \overleftrightarrow{EC} $
c) plane M
d) A, B, D
e) E

#### Problem 3:
a) Y, Z, T
b) plane XYZ (or plane XYP, etc.) — not plane R
c) $ \overline{YZ} $ — only if plane R is XYZ and plane ZYT is YZT
d) 5
e) 3 — but only if W is a point on 3 planes (likely typo)

#### Problem 4:
a) E
b) plane TFG
c) collinear
d) 1 (not 0)

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🟩 Final Answer:

The student did well overall, but made two key errors:
1. Problem 3b: Plane $ R $ does not contain $ X $ — so incorrect.
2. Problem 4d: Planes $ P $ and $ Q $ intersect in one line, not zero.

✔ Corrected answer for 4d: 1

Let me know if you'd like a version of the corrected worksheet!