The side AB measures option 2. [tex]\sqrt{20}}[/tex] units long.
Step-by-step explanation:
Step 1:
The coordinates of the given triangle ABC are A (4, 5), B (2, 1), and C (4, 1).
The sides of the triangle are AB, BC, and CA. We need to determine the length of AB.
To calculate the distance between two points, we use the formula [tex]d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}.[/tex]
where ([tex]x_{1},y_{1}[/tex]) are the coordinates of the first point and ([tex]x_{2},y_{2}[/tex]) are the coordinates of the second point.
Step 2:
For A (4, 5) and B (2, 1), ([tex]x_{1},y_{1}[/tex]) = (4, 5) and ([tex]x_{2},y_{2}[/tex]) = (2, 1). Substituting these values in the distance formula, we get
[tex]d=\sqrt{\left(2-4\right)^{2}+\left(1-5}\right)^{2}} = \sqrt{\left(2\right)^{2}+\left(4}\right)^{2}}=\sqrt{20}}.[/tex]
So the side AB measures [tex]\sqrt{20}}[/tex] units long which is the second option.
