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Chapter 2: Problem 21

How many lines can be found that contain (a) one given point (b) two given points (c) three given points?

Short Answer

Expert verified
(a) An infinite number of lines can pass through one given point. (b) There is only one line that can pass through two given points. (c) If the three points are collinear, only one line can pass through them; if not, there are three distinct lines for each pair of points.

Step by step solution

01

Part (a) - One Given Point

A line can be drawn through any point in an infinite number of directions. So, an infinite number of lines can pass through a single point.
02

Part (b) - Two Given Points

In geometry, two distinct points always define a unique line. Hence there is only one line that can pass through two given points.
03

Part (c) - Three Given Points

For three points, it depends on their arrangement. If the three points lie on the same line (are collinear) then only a single line can pass through them. However, if the points are not collinear, in other words, they are randomly arranged, each pair of points forms a unique line. Hence there would be 3 distinct lines for the 3 pairs: point 1 to point 2, point 2 to point 3, and point 3 to point 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Collinear Points
In geometry, points are considered collinear if they all lie on a single straight line. This simply means you can draw one straight line through all of them. Collinearity is an important concept as it determines whether three or more points can share a single geometric line.
  • If points are collinear, you can draw just one line that contains all of them.
  • To check if points are collinear, observe if they align straightly without deviation.
  • If they're not collinear, they end up forming a shape, like a triangle, instead.
Remember that non-collinear points are essential for forming shapes with area, while collinear points simply extend along one dimension. Understanding whether points are collinear helps with determining how they can be connected in geometric constructions.
Unique Line Through Two Points
A fundamental principle in geometry is that two distinct points uniquely define a line. This means that no matter where you are in space, if you have two separate points, there is exactly one straight line that passes through both. It's a reliable rule that simplifies lots of geometric work.
  • Think of drawing a line through any two points on a piece of paper 鈥� you can't draw a different line that still goes through those same points without it overlapping the first.
  • This rule ensures consistency in geometry, allowing us to construct and understand various figures based on this straightforward idea.
  • Knowing this rule, it becomes easier to predict the outcomes of geometric operations involving lines and points.
With this concept, two points set the path for just one line, allowing for precise and predictable geometric constructions.
Infinite Lines Through a Point
Through any given point, you can draw an infinite number of lines. This idea is foundational in geometry to help illustrate concepts relating to direction and dimension. When you have just one single point, imagine it as a central location from which countless lines radiate outwards in all directions, each representing a possible path.
  • No matter where you are, you can always imagine new lines starting at the same point that extend to unexplored directions.
  • These indefinite possibilities highlight the notion of a point being connected tomorrow to potentially infinite directions, each heading towards infinity.
  • This concept underscores the freedom and flexibility in geometric constructions starting from a single point.
Understanding this infinity of lines through a point is vital for grasping more complex geometric shapes and forms later on, as it allows us to explore endless possibilities.

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