Let's look at the three points and below.
As indicated in the diagram, the distances between the points are
Can we be sure that the points are collinear? To answer this, we use the segment addition postulate.
The postulate states the following:
Three points are collinear if and only if the largest distance between two of them is equal to the sum of the other two distances.
Here, the largest distance is so the points and are collinear if and only if
Substituting the given information and we get
This is a true statement, so the three points and are indeed collinear.
Note: The postulate works in both directions:
If the points and are collinear then
If then the points and are collinear.
Given two of the three distances between three collinear points, we can calculate the third distance using the segment addition postulate.
To demonstrate, consider the collinear points and below.
We are given two of the distances:
We can calculate the third distance, using the segment addition postulate.
First, we state the equation and substitute the known quantities and as follows:
Now, we can solve for by subtracting from both sides of the equation:
Therefore, the distance between the points and is
Let and be three collinear points such that is between and If and , find the length of .
The segment addition postulate states that
Substituting the known values into the formula above, we obtain:
Therefore, .
Let $P$, $Q$ and $R$ be three collinear points such that $Q$ is between $P$ and $R.$ If $PR = 59 \text{ cm}$ and $QR = 34 \text{ cm}$, find the length of $\overline{PQ}.$
|
a
|
$30 \text{ cm}$ |
|
b
|
$25 \text{ cm}$ |
|
c
|
$21 \text{ cm}$ |
|
d
|
$26 \text{ cm}$ |
|
e
|
$23 \text{ cm}$ |
Let $P$, $Q$ and $R$ be three collinear points such that $Q$ is between $P$ and $R.$ If $PQ = 8 \text{ cm}$ and $QR = 12 \text{ cm}$, find the length of $\overline{PR}.$
|
a
|
$18 \text{ cm}$ |
|
b
|
$16 \text{ cm}$ |
|
c
|
$20 \text{ cm}$ |
|
d
|
$21 \text{ cm}$ |
|
e
|
$22 \text{ cm}$ |
Given the diagram shown below, what is the measure of
The segment addition postulate can be applied to a set of more than three collinear points. In the given situation, it states that
Substituting the known values into the equation above, we obtain:
Given the diagram shown above, what is the measure of $\overline{BC}?$
|
a
|
$19$ |
|
b
|
$13$ |
|
c
|
$15$ |
|
d
|
$12$ |
|
e
|
$9$ |
Given the diagram shown above, what is the measure of $\overline{BC}?$
|
a
|
$11$ |
|
b
|
$12$ |
|
c
|
$13$ |
|
d
|
$8$ |
|
e
|
$10$ |
Given the diagram shown below, what is the value of
By the segment addition postulate, we have
To solve for we can substitute the given information
and then solve the resulting equation:
Given the diagram shown above, what is the value of $x?$
|
a
|
$8$ |
|
b
|
$3$ |
|
c
|
$6$ |
|
d
|
$9$ |
|
e
|
$7$ |
Given the diagram shown above, what is the value of $x?$
|
a
|
$7$ |
|
b
|
$5$ |
|
c
|
$6$ |
|
d
|
$3$ |
|
e
|
$4$ |
In the diagram below, the length of the line segment is and is the midpoint of Express the length of in terms of
First, note that according to the segment addition postulate, we have which, in turn, gives
Now, since is the midpoint of , we have that
Thus, we obtain:
In the diagram above, the length of the line segment $\overline{AB}$ is $5d+6,$ and $AD = 11d+12.$ If $C$ is the midpoint of $\overline{BD},$ find the length of $\overline{CD}.$
|
a
|
$6d+3$ |
|
b
|
$6d+5$ |
|
c
|
$3d+6$ |
|
d
|
$3d+3$ |
|
e
|
$2d+5$ |
In the diagram shown above, $B$ is the midpoint of $\overline{AD}.$ Find $BC$, if $AD = 20 \, \text{cm}$ and $CD=3 \, \text{cm}.$
|
a
|
$7 \, \text{cm}$ |
|
b
|
$5 \, \text{cm}$ |
|
c
|
$6 \, \text{cm}$ |
|
d
|
$4 \, \text{cm}$ |
|
e
|
$8 \, \text{cm}$ |