Answer: The correct option is (C) a right triangle.
Step-by-step explanation: The given co-ordinates of a polygon are A(2, 1), B(7, 1), C(2, 4).
We are to select the correct type of the polygon.
Since we have the co-ordinates of the three points A, B and C, so the polygon will surely be a triangle.
Now, the lengths of the three sides AB, BC and CA of ΔABC can be calculated using distance formula as follows :
[tex]AB=\sqrt{(7-2)^2+(1-1)^2}=\sqrt{25+0}=\sqrt{25}=5,\\\\BC=\sqrt{(2-7)^2+(4-1)^2}=\sqrt{25+9}=\sqrt{34},\\\\CA=\sqrt{(2-2)^2+(1-4)^2}=\sqrt{0+9}=\sqrt{9}=3.[/tex]
We have
[tex]AB^2+CA^2=5^2+3^2=25+9=34,\\\\BC^2=(\sqrt{34})^2=34.[/tex]
This implies that
[tex]AB^2+CA^2=BC^2,[/tex]
which is the Pythagorean triple. Hence, triangle ABC is a right-angled triangle.
Thus, (C) is the correct option.
