Introduction to Geometry (Level 5)
In this video we are going to go over basic examples that ask us to find the intersection
and union of various geometric figures. Before we jump into some examples lets refresh our
memory and talk about sets. Say I have two distinct sets let’s call them Set A and
Set B, recall that a set is a collection of objects, these objects are referred to as
the elements of the set. In this example the numbers 1, 2, 3, 4, 5, and 6 are the elements of set
A, likewise the numbers 2, 4, 6, 8,10, and 12 are the elements of set B. When dealing with sets
we are often asked to find the union or intersection between two or more sets.
If we are asked to find the union between set A and B we would denote it as follows:
A union B, this symbol is pretty easy to remember since it resembles the letter U for union.
In essence, the union between two sets consists of: {all elements in A or in B or in both}.
You can think of a union as the process of merging the elements of both sets together.
In this example A union B will be equal to a new set that contains every element that
is in either set A or set B including those elements that are common to both set A and B.
In contrast, if we are asked to find the intersection between set A and B we would denote it as
follows: A intersection B. This symbol looks like an upside down U and it sort of looks
like the capital letter A if you use your imagination, you can imagine the capital letter
A will remind you of the word “and” since the intersection of two sets creates a new
set containing the elements in both A and B. In this example A intersected B would equal
to the set containing the elements {2,4 and 6}. Since these elements are in both A and
B, graphically you can see the reason why this is called the intersection between two
sets its literally an intersection or overlapping of common elements.
The final concept we need to be familiar with is the idea of an empty set, let’s take Set A and Set B and remove the elements common to both sets. Now if we are asked to find A intersection
B we would see that there are no common elements between set A and B, in this case we would
obtain what is referred to as an empty set, and we use this symbol to denote the empty set, an empty set is special in the sense that it contains no elements. Alright now that we are refreshed and familiar
with the concepts associated with sets lets go ahead and try the first example.
Use the following figure to answer the following questions.
How many lines are shown? Name these lines. By looking at the figure we can see that there
are two distinct lines. So the answer to the first question is 2. Now the second question
is asking us to name them, let’s focus on the line that contains points D, B and E,
this line can be named by using two points so we can name this line as line DE or line DB or line BD or line or line BE or line EB or line ED, notice that this line is labeled with the lower case letter m, so we can also
name this line as line m. In the same manner, the line that contains points A, B and C can
be name as line AB or line AC or line BA or line BC or line CB or line CA. Remember lines can be named by denoting
two points located on the line or by using a lower case letter whenever possible. Alright, let’s
try the second example. Where do line AC and line DE intersect?
Using set notation this problem translates to the following expression: line AC intersection line DE. Looking at the lines we see that they intersect at a common point in this case point B. So line AC intersection
line DE is equal to point B. Let’s try the next example. Where does ray AC intersect line BC?
Using set notation this problem translates to the following expression. Recall that you
can think of a ray as being the set of all points that starts at one end point in this
case point A and goes in the direction of another point in this case point C, in the
same manner a line is the set of all points that extends infinitely in both directions
so the following points are located on line BC, for this problems we are looking for the
points that both ray AC and line BC have in common hence the intersection, in this case
the common points start intersecting at point A and extend infinitely in the direction of
point C, therefore the intersection of ray AC and line BC is equal to ray AC. Alright let’s
try the next example. What is the union of ray BA and ray BD?
Using set notation this problem translates to the following expression. First let’s
locate each of the rays, ray BA is located here and ray BD is located here, you can think of the union as the merger between these two rays, we want to include all the points on ray BA
or BD or in both. Recall that an angle is formed when two rays have a common point of
intersection called the vertex this is essentially what happens when we form the union of these
two rays, therefore the union of ray BA and ray BD is equal to angle ABD or angle DBA. Alright let’s
move along to the next example. Find the intersection of ray AB with ray CA.
Using set notation this problem translates to the following expression. Once again let’s
identify the set of all points located on ray AB, then we need to identify the set of all points
on ray CA, now it’s just matter of finding where the set of points of these two rays
intersect, we see that the set of points on both rays start intersecting at point A and
end at point C, in essence the set of points intersect along a common line segment in this
case line segment AC. So the intersection of these two rays is equal to line segment
AC. Alright, let’s end the video with the final example. Find the intersection of triangle ABC and
ray DE. Using set notation this problem translates
to the following expression. The set of all points located on triangle ABC are represented
as follows, and the set of all points located on ray DE are represented as follows, for
this problem we are looking for the intersection of these two set of points. Notice that both
sets intersect at a common point in this case they intersect at point F, so the intersection
of triangle ABC and ray DE is the set containing point F.
Alright in our next video we will continue covering slightly more challenging examples involving the union and intersection of geometric figures.