Chapter 6 tools of geometry
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Points are the basic elements in geometry with no properties other than location. Lines are idealized representations of objects with negligible width and depth. A plane is a flat, two-dimensional surface that can exist independently or as a subspace of a higher dimension. Collinear points all lie on the same line, and coplanar points all lie in the same plane. A line segment connects two endpoints and contains all intermediate points, while a ray has one endpoint and extends infinitely in one direction. The midpoint of a line segment is the point halfway between its two endpoints.
![CHAPTER 7 TOOLS OF GEOMETRY
PPooiinntt
In geometry, topology, and related branches of mathematics, a spatial point is a primitive notion upon which other
concepts may be defined. Being a primitive notion means that they have no properties other than those that are
derived from the axioms of the formal system in which they are used, i.e., they do not have volume, area, length, or
any other higher-dimensional attribute. In Euclidean geometry, a common interpretation is that the concept of a point
is meant to capture the notion of an object, with no properties, in a unique location in Euclidean space. In branches of
mathematics dealing with set theory, an element is sometimes referred to as a point.
LLiinnee
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with
negligible width and depth. Lines are an idealization of such objects. Until the seventeenth century, lines were
defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any
width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving
some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its
points"[1]
Euclid described a line as "breadthless length", and introduced several postulates as basic unprovable properties
from which he constructed the geometry, which is now called Euclidean geometry to avoid confusion with other
geometries which have been introduced since the end of nineteenth century (such as non-Euclidean geometry,
projective geometry, and affine geometry).
PPllaannee
In mathematics, a plane is a flat, two-dimensionalsurface. A plane is the two dimensional analogue of a point (zero-
dimensions), a line (one-dimension) and a solid (three-dimensions). Planes can arise as subspaces of some higher
dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in
the setting of Euclidean geometry.
When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the
whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are
performed in a two-dimensional space, or in other words, in the plane.
What are collinear and coplanar points?
Collinear points are all in the same line (linear means line).
When three or more points lie on a straight line.
(Two points make a line so would always be collinear.)
Coplanar points are all in the same plane.
Segments and Rays
A line segment has two endpoints. It contains these endpoints and all the points of the line between them. You can
measure the length of a segment, but not of a line.
A segment is named by its two endpoints, for example, .
A ray is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the
length of a ray.](https://image.slidesharecdn.com/chapter6toolsofgeometry-140321211132-phpapp02/85/Chapter-6-tools-of-geometry-1-320.jpg)
![A ray is named using its endpoint first, and then any other point on the ray (for example, ).
Midpoint of a Line Segment
Definition of Midpoint of a Line Segment
Midpoint is the point on a line segment and is halfway between the endpoints of the line segment.
Examples of Midpoint of a Line Segment
In the figure shown, C is the midpoint of the line segment AB.
That is, AC = CB.
More about Midpoint of a Line Segment
Midpoint Formula: The formula to find the midpoint of a line segment with endpoints
is .
Solved Example on Midpoint of a Line Segment
Find the midpoint of the line segment whose endpoints are (8, - 16) and
(4, - 4).
Choices:
A. (6, 10)
B. (10, 6)
C. (6, - 10)
D. (- 6, 10)
Correct Answer: C
Solution:
Step 1: The endpoints of the line segment has coordinates (8, - 16) and
(4, - 4).
Step 2: Midpoint of the Line Segment with endpoints is .
[Use the midpoint formula.]
Step 3: Midpoint = . [Substitute the values.]
Step 4: [Simplify.]
Step 5: = (6, - 10) [Write the fractions in the simplest form.]
Step 6: So, the midpoint of the line segment is (6, - 10).](https://image.slidesharecdn.com/chapter6toolsofgeometry-140321211132-phpapp02/85/Chapter-6-tools-of-geometry-2-320.jpg)
Chapter 6 tools of geometry
- 1. CHAPTER 7 TOOLS OF GEOMETRY PPooiinntt In geometry, topology, and related branches of mathematics, a spatial point is a primitive notion upon which other concepts may be defined. Being a primitive notion means that they have no properties other than those that are derived from the axioms of the formal system in which they are used, i.e., they do not have volume, area, length, or any other higher-dimensional attribute. In Euclidean geometry, a common interpretation is that the concept of a point is meant to capture the notion of an object, with no properties, in a unique location in Euclidean space. In branches of mathematics dealing with set theory, an element is sometimes referred to as a point. LLiinnee The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. Until the seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its points"[1] Euclid described a line as "breadthless length", and introduced several postulates as basic unprovable properties from which he constructed the geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of nineteenth century (such as non-Euclidean geometry, projective geometry, and affine geometry). PPllaannee In mathematics, a plane is a flat, two-dimensionalsurface. A plane is the two dimensional analogue of a point (zero- dimensions), a line (one-dimension) and a solid (three-dimensions). Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry. When working exclusively in two-dimensional Euclidean space, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane. What are collinear and coplanar points? Collinear points are all in the same line (linear means line). When three or more points lie on a straight line. (Two points make a line so would always be collinear.) Coplanar points are all in the same plane. Segments and Rays A line segment has two endpoints. It contains these endpoints and all the points of the line between them. You can measure the length of a segment, but not of a line. A segment is named by its two endpoints, for example, . A ray is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray.
- 2. A ray is named using its endpoint first, and then any other point on the ray (for example, ). Midpoint of a Line Segment Definition of Midpoint of a Line Segment Midpoint is the point on a line segment and is halfway between the endpoints of the line segment. Examples of Midpoint of a Line Segment In the figure shown, C is the midpoint of the line segment AB. That is, AC = CB. More about Midpoint of a Line Segment Midpoint Formula: The formula to find the midpoint of a line segment with endpoints is . Solved Example on Midpoint of a Line Segment Find the midpoint of the line segment whose endpoints are (8, - 16) and (4, - 4). Choices: A. (6, 10) B. (10, 6) C. (6, - 10) D. (- 6, 10) Correct Answer: C Solution: Step 1: The endpoints of the line segment has coordinates (8, - 16) and (4, - 4). Step 2: Midpoint of the Line Segment with endpoints is . [Use the midpoint formula.] Step 3: Midpoint = . [Substitute the values.] Step 4: [Simplify.] Step 5: = (6, - 10) [Write the fractions in the simplest form.] Step 6: So, the midpoint of the line segment is (6, - 10).