Collinearity, Betweenness, and Assumptions, Level 1.
In the following series of videos we will go over the concepts of collinearity, betweenness
of points, triangle inequality, and assumptions from diagrams.
Let’s start with the concept of collinearity.
It is often useful to know that a group of points lie on the same line.
Points that lie on the same line are called collinear.
The word Co is latin for “together” and linearis which means “belonging to a line”.
Points that do not lie on the same line are called non-collinear.
In the diagram shown points A, B and C are collinear points since they all lie in the
same line.
Points D, E and F are non-collinear points since they do not lie in the same line.
We want to be able to identify collinear and non-collinear points when we are solving problems
in geometry.
In the following diagram points R, S and T are collinear points.
Points P, O, and X are also collinear.
Notice that points M, O, X and Y are noncollinear.
Alright let’s move along and talk about Betweenness of points.
In order for us to say that a point is between two other points, all three of the points
must be collinear.
For example in the following diagram point T is between point A and point R since all
three points are collinear.
On the other hand we cannot say that point O is between point X and point Y since all
three points are not in the same line.
Notice that 3 points that are not collinear form the vertices of a triangle.
For any three points, there are only two possibilities.
The first possibility is that all three points are collinear.
That is to say that one point is between the other two points.
When three points are collinear we can break apart this line into 3 distinct line segments
each with their own length or measurement.
Two of the lengths add up to the third length.
For example the following points are collinear and point B is between point A and point C.
If the length of line segment AB is 10 and the length of line segment BC is 15 then the
length of line segment AC will be equal to the sum of the length of line segment AB and
the length of line segment BC in this case it will be equal to 25.
The second possibility is that all three points are noncollinear.
In other words the three points determine a triangle.
If we keep the length of line segment AC intact and move point B so that it is not between
point A and point C we would form a triangle.
Something interesting happens with the lengths of the triangle; with the points arranged
this way the length of line segment AB measures 10 and the length of line segment BC measures
16.
If we were to add the length of line segment AB and the length of line segment BC we would
obtain a length that is greater than the length of line segment AC.
When the lines were collinear the sum of the lengths of the two smaller line segments was
equal to the length of the larger line segment that included all three points.
This is not the case with non-collinear points.
In this case the sum of the lengths of two sides of the triangle is greater than the
length of the third side.
This makes sense since the shortest distance between two points in a plane is a straight
line.
The total distance covered going from point A to point C will always be longer if you
have to 'detour' via point B as oppose to going in a straight line directly from point
A to point C.
This is an example of an important characteristic of triangles in this case the sum of the lengths
of any two sides of a triangle is always greater than the length of the third side.
Said another way any side of a triangle is always shorter than the sum of the other two
sides.
This fact generates 3 separate inequalities.
In the first inequality the sum of the length of segment AB plus the length of segment BC
is greater than the length of segment AC.
In the second inequality the sum of the length of segment AB plus the length of segment AC
is greater than the length of segment BC, and in the third inequality the sum of the
length of segment BC and the length of segment AC is greater than the length of segment AB.
These 3 inequalities must always be obeyed for any triangle in a plane.
Alright let’s end the video by going over assumptions from diagrams.
It is important to understand what you should and should not assume when you look at a diagram.
In general one interprets a diagram by assuming the following: straight lines and straight
angles, collinearity of points, betweenness of points and relative positions of points.
You should not assume the following: right angles, this is one of the most common mistakes
that many students make.
Never ever assume that an angle is a right angle, just because it looks like a right
angle does not mean that it is a right angle.
The second most common mistake is to assume congruent line segments and congruent angles,
again just because two line segments or two angles look congruent does not mean that they
are congruent.
Lastly you should never assume the relative sizes of segments and angles.
Just because a segment or angle looks like it is twice as large as another segment or
angle does not mean that they are twice as large.
One of the reasons why students make these incorrect assumptions is because of wishful
thinking, if these assumptions are made it will make the problem easier to solve.
So when you are interpreting diagrams try to fight the urge to make an assumption just
because the problem will be easier to solve if those assumptions were true.
There are occasional exceptions and most textbooks and test questions will have a note indicating
that the diagram is not drawn to scale.
If this is the case be extremely careful with making incorrect assumptions.
One has to prove that these assumptions are true and we will go over how to do this in
a much later video.
Let’s go over an example to illustrate the appropriate way to interpret a diagram.
The following are some of the many valid interpretations.
One can assume from the diagram that line ACD and line BCE are straight lines.
Angle BCE and angle ACD are straight angles.
Points C, D, and E and points C, A, and B are non-collinear.
Point C is between point B and point E. Point E is to the right of point A and point B is
to the left of point D. Now, you should not assume that angle BAC
or angle CDE are right angles.
Nor should you assume that segment CD is congruent to segment DE or that segment AC is congruent
to segment AB.
You also cannot assume that angle B is congruent to angle E. You should not assume that angle
CDE is an obtuse angle.
You should not assume that segment BC is longer than segment CE.
Alright, make sure you understand how to interpret a diagram because it will be extremely important
that you know what to and what not to assume from a given diagram.
In our next video we will go over various examples that make use of these new concepts.