Introduction to Geometry (Level 6) In this video we are going to use the following
figure to find the union and intersection of various lines, line segments, and rays.
Recall that the symbol for denoting the union of two sets is this u shape symbol it is essentially
a U for union. The symbol for the intersection of two sets is an upside down U. It is important
to understand that, when we are dealing with lines, line segments and rays, a set will consist
of all the points that a particular line, line segment or ray is composed of. For example
you can think of a line as the set of all points that extend infinitely in both directions,
a line segment is the set of all points that starts at one point and ends at another point,
and a ray is the set of all points that starts at one point and extends infinity in one direction.
With that said lets go ahead and start with the first example.
Use the following figure to find the following unions and intersections.
In this first problem they are asking us to find the intersection of line segment AD and
DC. If we think of line segment AD as the set of all points that forms this line we
would obtain the following points, in the same manner the set of all points that form
line segment DC will consist of the following points. Since we are trying to find the intersection
of these two set of points we see that both lines contain one point in common in this
case point D. So the intersection of line segment AD and DC is the set containing only point D.
Now let’s try the second example In this problem they are asking us to find
the union of ray EC and ray EA, notice that the diagram does not contain any arrows, never
the less we can either imagine or add the arrows in the diagram when needed. For this example
we are referring to ray EC which is the set of all points located here even the points
that extend infinitely in this direction, in the same manner the set of all points of
ray EA are located here again even the points that extend infinitely in this direction.
Since we are asked to find the union of these two set of points we essentially want to unite
or merge all the points from both rays forming a new set. This new set will form a line segment which
can be denoted as line segment AC or line segment CA. We can also argue that the union
of these two rays also forms an angle in this case we form a straight angle specifically
angle AEC or angle CEA with the vertex located at point E, straight angles will be
discussed more fully in a later video. For now, either the line segment or angles are
acceptable answers for this problem. Alright, let’s go ahead and try the next example.
Here we are asked to find the intersection of line AC and line BD, the set of all points
forming line AC are represented as follows and that of line BD are represented as follows,
since we are looking for the intersection of these two lines we see that they have point
E in common so the intersection of these two lines is the set that contains point E.
Alright, let’s move along to the next example. Here we need to find the intersection of line
segment BC and line segment AD, the set of all points that are associated with line segment
BC are represented as follows and those associated with line segment AD are represented as follows,
notice that none of the points of either line segment BC or line segment AD intersect,
this means that there are no points that both line segments have in common, this results in an empty set,
which is represented by the following symbol. Ok, let’s try the next example.
Here we need to find the intersection of ray AC and ray EC, here we have to be careful
since the way we denote rays is going to be key in this problem, the first letter on a
ray represents the end point and the second point, is in the direction in which the ray
infinitely extends towards. The set of all points of ray AC is represented as follows,
and the set of points of ray EC is represented as follows, for this problem we are looking
for the intersection of these two rays, this occurs in the following region, notice that
it forms a ray with end point located at point E and extends infinitely towards point C,
so the intersection of these two set of points is the set of all points that forms ray EC.
Alright, let’s proceed to the next example.
Here we need to find the union of ray DA and ray DC. The set of all points represented
by ray DA consist of the following points, and the set of all points represented by ray
DC consist of the following points, since we are looking for the union of these
two rays, we see that we essentially form angle ADC or angle CDA with vertex located at
point D. Remember when we are asked to find the union of two sets we want to merge all
the elements together in this case all the points that are located on ray DA, or ray DC or in both.
Alright, let's end the video with the final example.
Here we need to find the union of line segment EC, line segment CD and line segment DE.
Notice that we have three distinct sets the concepts and ideas are the same regardless of the number
of sets. So don’t panic if you get a problem that contains multiple sets. Alright, the
set of all points represented by line segment EC consist of the following points the set of all
points for line segment CD are represented as follows, and the set of points, of line
segment DE consist of the following points. Since we are trying to find the union of all
of these points we see that by merging all of these points together we create a new set,
we create a new set of points that represents a triangle.
Specifically triangle ECD or written in other ways as:
triangle CDE or triangle DEC or triangle DCE or triangle CED or triangle EDC.
As long as you label the points forming the line segment or sides of the triangle clock wise or counter clockwise.
Alright in our next and final video on this
section, we are going to go over more challenging problems that ask us to name and find the
union and intersection of geometric figures.