To match the given lengths with the corresponding segments, you have to calculate the distance between the endpoints of each line segment.
First, using the graph, you have to determine the coordinates of each point:
A(5,6)
B(8,-6)
C(-4,-4)
D(-7,2)
Then, using the following formula you can calculate the length of each segment:
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Where
(x₁,y₁) represent the coordinates of the endpoint with the smallest x-coordinate
(x₂,y₂) represent the coordinates of the endpoint with the largest x-coordinate
Segment AC
The coordinates of the endpoints are: A(5,6) and C(-4,-4)
A has the largest x-coordinate, so you have to use it as (x₂,y₂)
C has the smallest x-coordinate, so you have to use it as (x₁,y₁)
Replace the coordinates on the formula and calculate the length of the segment:
[tex]\begin{gathered} AC=\sqrt[]{(5-(-4))^2+(6-(-4))^2} \\ AC=\sqrt[]{(5+4)^2+(6+4)^2} \\ AC=\sqrt[]{9^2+10^2} \\ AC=\sqrt[]{81+100} \\ AC=\sqrt[]{181} \end{gathered}[/tex]The length of segment AC is √181 units.
Segment AB
The endpoints of this segment have the coordinates A(5,6) and B(8,-6)
Use B as (x₂,y₂) since it has the largest x-coordinate.
Use A as (x₁,y₁) since it has the smallest x-coordinate.
Replace them on the formula and calculate the length of segment AB
[tex]\begin{gathered} AB=\sqrt[]{(8-5)^2+((-6)-6)^2} \\ AB=\sqrt[]{(3)^2+(-12)^2}^{} \\ AB=\sqrt[]{9+144} \\ AB=\sqrt[]{153} \end{gathered}[/tex]The length of segment AB is √153 units.
Segment CD
The endpoints of this segment are C(-4,-4) and D(-7,2).
Use C as (x₂,y₂) since it has the largest x-coordinate.
Use D as (x₁,y₁) since it has the smallest x-coordinate.
Replace the coordinates on the formula and calculate the length of the segment:
[tex]\begin{gathered} CD=\sqrt[]{((-4)-(-7))^2+((-4)-2)^2} \\ CD=\sqrt[]{(3)^2+(-6)^2} \\ CD=\sqrt[]{9+36} \\ CD=\sqrt[]{45} \end{gathered}[/tex]The length of segment CD is √45 units.
Segment BD
The endpoints of this segment are B(8,-6) and D(-7,2).
Use B as (x₂,y₂) since it has the largest x-coordinate.
Use D as (x₁,y₁) since it has the smallest x-coordinate.
[tex]\begin{gathered} BD=\sqrt[]{(8-(-7))^2+((-6)-2)^2} \\ BD=\sqrt[]{(8+7)^2+(-8)^2} \\ BD=\sqrt[]{(15)^2+(-8)^2} \\ BD=\sqrt[]{225+64} \\ BD=\sqrt[]{289} \\ BD=17 \end{gathered}[/tex]The length of segment BD is 17 units.