Division of Segments and Angles.
Level 4.
In this video we will go over more challenging example involving midpoints, bisectors and
trisectors.
Let's go ahead and jump straight into the first example.
In the diagram, point X is the midpoint of segment VZ, the length of segment VW equals
5, and the length of segment VY equals 20.
Find the coordinates of point W, X, and Y.
In this problem we are provided with a diagram of a number line with points located at particular
coordinates along with the measurement of two line segments.
We are asked to determine the coordinates of three of these points.
Alright let's start the problem by marking the diagram with the given information.
We know that point X is the midpoint of segment VZ this means that segment VX and segment
XZ are congruent so the segments lengths are also equal to one another.
We also know that the length of segment VW is equal to 5 and the length of segment VY
is equal to 20.
With this information we can determine the coordinates of each of these three points.
For point W we know that its coordinate is going to be 5 units to the right of point
V, which has a coordinate of negative 12.
Moving 5 units to the right from negative 12 we land on negative 7.
In a similar fashion the coordinate of point Y is going to be 20 units to the right of
point V in this case point Y will have a coordinate of 8.
Lastly the coordinate of point X can be determined by finding the length of segment VZ and taking
half this length.
To find the length of segment VZ we can compute the absolute difference of the segment's
endpoint coordinates.
Doing that we obtain 34.
We now take half this length, which is equal to 17 and use it to find the coordinate of
point X by moving either 17 units to the right of point V or 17 units to the left of point
Z doing that we obtain 5 for the coordinate of point X and this is our final answer.
Alright let's take a look at the next example.
Segment EH is divided by point F and point G in the ratio 5 to 3 to 2 from left to right.
If the length of segment EH is 30, find the length of segment FG and name the midpoint
of segment EH.
In this problem we are provided with a line segment that is divided into 3 smaller segments.
We are given the length of one of these segments as well as the ratio of the lengths between
these 3 segments.
We are asked to determine the length of one of the segments and identify the midpoint
of the largest segment.
For this type of problem we are going to make use of the geometric relations that are provided
to us and use them to set up an algebraic relation.
We know from the problem that the length of the 3 segments are in the ratio of 5 to 3
to 2 since we do not know the actual length of each of the segments we are going to use
a variable and let the length of segment EF be equal to 5x, the length of segment FG equal
to 3x, and the length of segment GH equal to 2x.
This way the algebraic expressions are consistent with the ratios given to us.
Next we notice that point E, F, G and H are collinear this means that the length of segment
EF plus the length of segment FG plus the length of segment GH will be equal to the
length of segment EH.
Using this geometric relation we can go ahead and substitute the algebraic expressions for
the length of each segment as follows.
Now it is just a matter of solving for x so we collect like terms and divide both sides
of the equation by 10 doing that we obtain x equals 3.
With this value of x we can now determine the length of each line segment.
In this case line segment EF is equal to 15, line segment FG is equal to 9 and line segment
GH is equal to 6.
Notice that line segment EF has the same length as segment FH they are both equal to 15 this
means that point F is the midpoint of segment EH since it divides segment EH into two congruent
segments and this is our final answer.
Alright let's try the next example.
Given that the length of segment OM is equal to x plus 8, the length of segment MP is equal
to 2x minus 6, and the length of segment OP is equal to 44.
Determine if point M is the midpoint of segment OP.
In this problem we are provided with a diagram of a line segment along with some algebraic
and numerical expressions which represent the lengths of two smaller line segments.
We are asked to determine if a point represents the midpoint of a line segment.
In order to determine if point M is the midpoint of line segment OP we are going to use the
definition of a midpoint along with the algebraic and numerical expressions provided in the
problem.
We know that the length of segment OP is equal to 44 so if point M is the midpoint of line
segment OP then it will bisect the segment into two congruent segments each measuring
half of 44 in this case each one will measure 22.
We can use this length and set it equal to the algebraic expressions representing segment
OM and MP.
We will then solve for x for each equation.
If we happen to get the same solution for both equations then we can conclude that point
M is the midpoint of line segment OP.
If we get a different solution for each equation then we can conclude that point M is not the
midpoint of line segment OP.
So solving for x on each equation we obtain x equals 14 for the first equation and x equals
14 for the second equation.
This means that point M is the midpoint of line segment OP, since the same value of x
satisfies both equations and this is our final answer.
Alright let's take a look at the next example.
Point B and point C are trisection points of segment AD, and the measurement of segment
AD equals 12.
Find the length of segment AB and segment AC.
In this problem we are provided with a diagram, the measurement of a line segment and the
fact that two points are trisection points of a line segment.
We are asked to determine the length of two smaller line segments.
We are first going to label the diagram with all the given information in this case the
length of segment AD is equal to 12.
We will also mark segment AB, segment BC and segment CD with single tick marks since point
B and point C are trisection points of segment AD.
Recall that trisection points split a line segment into three congruent segments.
Now that we have labeled the diagram we can go ahead and start solving for the length
of segment AB and segment AC.
We know that the length of segment AD is equal to 12.
We also know that segment AD is split into three congruent segments.
Each of the congruent segments has a measurement equal to twelve divided by 3 which is equal
to 4, this is also the measurement of segment AB.
The measurement of segment AC will be equal to twice the measurement of segment AB in
this case it will be equal to 8 and this is our final answer.
Let's end the video with the final example.
If the length of segment AB is equal to x plus 3 and the length of segment AE is equal
to 3x plus 6 find the length of segment AE.
In this problem we are provided with algebraic expressions representing the lengths of two
distinct line segments.
We are asked to find the length of one of these segments.
From the previous example we know that segment AB measures 4 so we can equate this value
with the variable expression x plus 3.
This will allow us to solve for the unknown variable x.
Doing that we obtain x equals 1.
Now that we know the value of x we can go ahead and substitute it into the algebraic
expression representing the length of segment AE.
Substituting this value for x and simplifying we obtain 9 for the measurement of segment
AE and this is our final answer.
Alright in our next video we will continue gong over more challenging examples involving
midpoints and segment bisectors.
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