Question 1192321: 1. Suppose you have n points, no three of which are collinear. How many lines contain two of these n points?
2. If no four of the n points are coplanar, how many planes contain three of the n points?
Hint: (for 3 and 4, generalize in a form of a formula)
3.Prove theorem 1.1.4. The steps in the proof are already given: you just have to supply the reasons for each step.
Theorem 1.1.4. If two lines intersect, then their union lies in exactly one plane.
Proof: Let and be two intersecting lines.
a. A ∩ B is a point p.
b. B contains a point q ≠ p.
c. There is a plane E, containing A and q.
d. E contains A ∪ B.
e. No other plane contains A ∪ B.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Certainly, let's break down these geometry problems:
**1. Number of Lines from n Points**
* **Concept:** To form a line, we need to select 2 points from the given 'n' points.
* **Calculation:** The number of ways to select 2 points from 'n' points is given by the combination formula:
* nC2 = n! / (2! * (n-2)!)
* where 'n!' denotes the factorial of n (n! = n * (n-1) * (n-2) * ... * 1)
**Therefore, the number of lines that can be formed from 'n' non-collinear points is nC2.**
**2. Number of Planes from n Points**
* **Concept:** To form a plane, we need to select 3 non-collinear points.
* **Calculation:** The number of ways to select 3 points from 'n' points is given by:
* nC3 = n! / (3! * (n-3)!)
**Therefore, the number of planes that can be formed from 'n' non-coplanar points is nC3.**
**3. Proof of Theorem 1.1.4**
* **a. A ∩ B is a point p.**
* **Reason:** By definition, intersecting lines share exactly one point in common.
* **b. B contains a point q ≠ p.**
* **Reason:** Since B is a line, it must contain at least two distinct points.
* **c. There is a plane E, containing A and q.**
* **Reason:** Two distinct points (p and q) determine a unique line. A line and a point not on that line uniquely determine a plane.
* **d. E contains A ∪ B.**
* **Reason:**
* E contains line A (by definition).
* E contains point q (by definition).
* Since E contains a point (p) and a line (A) that lies entirely within the plane, it must also contain all other points on that line (including all points on line B).
* **e. No other plane contains A ∪ B.**
* **Reason:** Two intersecting lines uniquely define a plane.
I hope this explanation is helpful! Let me know if you have any further questions.
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