Question

The coordinates A(2, 1), B(7, 1), C(2, 4) form what type of polygon?

A. an acute triangle
B. an obtuse triangle
C. a right triangle
D. an isosceles triangle

Answer 1

It appears to be a right triangle.

Answer 2

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 :

$AB=\sqrt{(7-2)^2+(1-1)^2}=\sqrt{25+0}=\sqrt{25}=5,\\\\BC=\sqrt{(2-7)^2+(4-1)^2}=\sqrt{25+9}=\sqrt{34},\\\\CA=\sqrt{(2-2)^2+(1-4)^2}=\sqrt{0+9}=\sqrt{9}=3.$

We have

$AB^2+CA^2=5^2+3^2=25+9=34,\\\\BC^2=(\sqrt{34})^2=34.$

This implies that

$AB^2+CA^2=BC^2,$

which is the Pythagorean triple. Hence, triangle ABC is a right-angled triangle.

Thus, (C) is the correct option.