Points, Lines, and Planes


Space is a 3-dimensional set of all points, lines, and planes.  A point is a location.

line contains a series of points that extend forever in opposite directions.  Points that lie on the same line are called collinear.  A line has no thickness or width.  There is exactly one line through any two points.  The line can be named by its italicized lower-case letter.  Also, any two points on the line can be used to name it.  A part of a line between two endpoints is called a line segment.

plane is a flat flat surface that contains points and lines.  Points that lie on the same plane are called coplanar.  The plane can be named by its italicized upper-case letter.  Or, it can be named using any three noncollinear points in the plane.

Angles

If two noncollinear rays have a common endpoint, then they form an angle.  The rays are the sides of the angle.  The common endpoint is the vertex.

right angle is an angle whose measure is 90 degrees.  An acute angle has measure less than 90 degrees.  An obtuse angle has a measure greater than 90 but less than 180.  Angles that have the same measure are congruent angles.  A ray that divides an angle into two congruent angles is called an angle bisector.

Adjacent angles are angles in the same plane that have a common vertex and a common side, but no common interior points .

Vertical angles are two nonadjacent angles formed by two intersecting lines.  A pair of adjacent angles whose noncommon sides are opposite rays is called a linear pair.

2-Dimensional Figures

A closed figure formed by a finite number of coplanar line segments is known as a polygon.  A polygon is named according to its number of sides.  A regularpolygon has congruent sides and congruent angles.  A polygon can be convex or concave.

The perimeter of a polygon is the sum of the lengths of all the sides of the polygon.  The circumference of a circle is the distance around the circle.  Thearea is the number of square units needed to cover a surface.

Midpoint and Distance in the Coordinate Plane

The midpoint of a line segment cuts the segment into two congruent halves.  If a segment has endpoints with coordinates (x1, y1) and (x2, y2), then we can find the coordinates of a midpoint by using the midpoint formula located on the left.

Watch the video below to see the Midpoint Formula in action.

Midpoint Formula in Reverse

If you are given the coordinates of the midpoint and only one endpoint of the line segment, and you’re asked to find the coordinates of the missing endpoint, you can use the Midpoint Formula in reverse.

Distance Formula

To find the distance between two points in the coordinate plane, we use the Distance Formula.  The Distance Formula is actually a modification of the Pythagorean Theorem that uses the change in y and the change in x as the legs of a right triangle.  The hypotenuse would represent the distance between the two points.


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