Collinearity, Betweenness, and Assumptions, level 3.
In this video we will go over 3 slightly more challenging examples illustrating how to use
the concepts of collinearity and triangle inequality.
Let's jump straight into the first example.
Given the length of segment WY is equal to 25 and the ratio of the length of segment
WX to the length of segment XY is 3 to 2.
Find the length of segment WX.
In this problem we are provided with a diagram along with some information describing the
line segments.
We are asked to find the length of segment WX.
Let's first label the diagram with all the given information before we start solving it.
We know that the length of segment WY is equal to 25.
We also know that the ratio of the length of segment WX to the length of segment XY
is 3 over 2.
In order to make this ratio useful for solving the problem we are going to rewrite the fraction
3 over 2 to an equivalent fraction by multiplying this ratio by x over x which is just equal
to 1 so multiplying the fraction 3 over 2 by 1 does not change the value of the fraction,
what it will do is rewrite this ratio into an equivalent ratio in terms of the variable
x doing that we obtain the following.
This ratio now contains algebraic relations that can be used with geometric relations.
In this case the length of segment WX is equal to 3x and the length of segment XY is equal
to 2x.
Now we need to determine the geometric relations of the problem.
Since point X is between point W and point Y we know that the sum of the lengths of segment
WX and segment XY must be equal to the length of segment WY in this case it is equal to
25.
Next we substitute the algebraic relations for the length of the smaller line segments
into this geometric relation doing that we obtain the following expression.
Now it is just a matter of simplifying and solving for x doing that we obtain x equals
5.
The final step is to find the length of segment WX so we substitute the value of x into the
algebraic expression doing that we obtain 15 and this is our final answer.
Alright let's try the next problem.
Find the value of y if the length of segment GE is equal to 3y, the length of segment GH
is equal to 7y minus 4 and the length of segment EH is equal to 24.
In this problem we are provided with a diagram and algebraic expressions for the lengths
of 3 line segments.
We are asked to solve for the variable y.
We know from the diagram that all 3 points are collinear so if we were to add the length
of segment GE and the length of segment EH it should equal to the length of segment GH.
Using this geometric relation we go ahead and substitute the algebraic expressions representing
the length of the line segments into this geometric relation, doing that we obtain the
following.
Now it is just a matter of solving for y so we collect like terms and carry out inverse
operations to solve for y doing that we obtain y equals 7 and this is our final answer.
Alright let's end the video with the final example.
Given the diagram of triangle ABC determine the following: the length of segment AC must
be smaller than what number?
The length of segment AC must be larger than what number?
In this problem we are provided with a diagram of a triangle along with the length of two
of its sides, we need to answer two distinct questions.
Both questions are asking us to determine the possible values that the length of segment
AC can have.
We can determine these values by making use of the triangle inequalities.
Recall that if 3 points are not collinear then they define a triangle.
An important characteristic of triangles is that the sum of the lengths of any two sides
of a triangle is always greater than the length of the third side.
So we can write 3 distinct inequalities that this triangle must obey.
The first inequality is the length of segment AB plus the length of segment BC is greater
than the length of segment AC.
The second inequality is the length of segment AC plus the length of segment AB is greater
than the length of segment BC, and the last inequality is the length of segment BC plus
the length of segment AC is greater than the length of segment AB.
Now that we have these 3 inequalities it is just a matter of substituting the known lengths.
In this problem we know that the length of segment AB is 6 and the length of segment
BC is 9.
The final step is to solve for the length of segment AC for all 3 inequalities doing
that we obtain the following.
The first inequality tells us that the length of segment AC has to be smaller than 15 so
this inequality answers the first question.
The second inequality tells us that the length of segment AC has to be greater than 3 and
this answers the second question.
Notice that the third inequality is not applicable since lengths can only be positive and not
negative so this inequality provides no useful information.
In the end the length of segment AC has to be greater than 3 and less than 15 to form
a triangle with the given side lengths.
Alright in our final video we will go over examples that involve angle measurements.
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