Part A:
The
parametric equations
![x=x_1+(x_2-x_1)t](https://tex.z-dn.net/?f=x%3Dx_1%2B%28x_2-x_1%29t)
and
![y=y_1+(y_2-y_1)t](https://tex.z-dn.net/?f=y%3Dy_1%2B%28y_2-y_1%29t)
where
![0\leq t\leq1](https://tex.z-dn.net/?f=0%5Cleq%20t%5Cleq1)
describes the line segment that joins the points
![(x_1,\ y_1)](https://tex.z-dn.net/?f=%28x_1%2C%5C%20y_1%29)
and
![(x_2,\ y_2)](https://tex.z-dn.net/?f=%28x_2%2C%5C%20y_2%29)
.
The parameterization of the line joining points A(1, 1) and B(5, 3) are given by substituting
![x_1=1,\ \ y_1=1,\ \ x_2=5,\ \ y_2=3](https://tex.z-dn.net/?f=x_1%3D1%2C%5C%20%5C%20y_1%3D1%2C%5C%20%5C%20x_2%3D5%2C%5C%20%5C%20y_2%3D3)
.
Thus, we have
x = 1 + (5 - 1)t = 1 + 4t
y = 1 + (3 - 1)t = 1 + 2t
Therefore, the parameterization of line joining points A(1, 1) and B(5, 3) are x = 1 + 4t and y = 1 + 2t.
Part B:
The
parametric equations
![x=x_1+(x_2-x_1)t](https://tex.z-dn.net/?f=x%3Dx_1%2B%28x_2-x_1%29t)
and
![y=y_1+(y_2-y_1)t](https://tex.z-dn.net/?f=y%3Dy_1%2B%28y_2-y_1%29t)
where
![0\leq t\leq1](https://tex.z-dn.net/?f=0%5Cleq%20t%5Cleq1)
describes the
line segment that joins the points
![(x_1,\ y_1)](https://tex.z-dn.net/?f=%28x_1%2C%5C%20y_1%29)
and
![(x_2,\ y_2)](https://tex.z-dn.net/?f=%28x_2%2C%5C%20y_2%29)
.
The parameterization of the line joining
points B(5, 3) and C(1, 7) are given by substituting
![x_1=5,\ \ y_1=3,\ \ x_2=1,\ \ y_2=7](https://tex.z-dn.net/?f=x_1%3D5%2C%5C%20%5C%20%0Ay_1%3D3%2C%5C%20%5C%20x_2%3D1%2C%5C%20%5C%20y_2%3D7)
.
Thus, we have
x = 5 + (1 - 5)t = 5 - 4t
y = 3 + (7 - 3)t = 3 + 4t
Therefore, the parameterization of line joining points B(5, 3) and C(1, 7) are x = 5 - 4t and y = 3 + 4t.
Part C:
The
parametric equations
![x=x_1+(x_2-x_1)t](https://tex.z-dn.net/?f=x%3Dx_1%2B%28x_2-x_1%29t)
and
![y=y_1+(y_2-y_1)t](https://tex.z-dn.net/?f=y%3Dy_1%2B%28y_2-y_1%29t)
where
![0\leq t\leq1](https://tex.z-dn.net/?f=0%5Cleq%20t%5Cleq1)
describes the
line segment that joins the points
![(x_1,\ y_1)](https://tex.z-dn.net/?f=%28x_1%2C%5C%20y_1%29)
and
![(x_2,\ y_2)](https://tex.z-dn.net/?f=%28x_2%2C%5C%20y_2%29)
.
The parameterization of the line joining
points C(1, 7) and A(1, 1) are given by substituting
![x_1=1,\ \ y_1=7,\ \ x_2=1,\ \ y_2=1](https://tex.z-dn.net/?f=x_1%3D1%2C%5C%20%5C%20%0Ay_1%3D7%2C%5C%20%5C%20x_2%3D1%2C%5C%20%5C%20y_2%3D1)
.
Thus, we have
x = 1 + (1 - 1)t = 1
y = 7 + (1 - 7)t = 7 - 6t
Therefore, the parameterization of line joining points A(1, 1) and B(5, 3) are x = 1 and y = 7 - 6t.