Question

The vertices of the triangle are (-2, 0), (2, 3) and (1, -3). Is the triangle equilateral, isosceles or scalene?

Answer 1

Hint: We know the definition of equilateral triangle, isosceles triangle and scalene triangle. In equilateral triangles all the sides are equal. In the isosceles triangle only two sides are equal. In the scalene triangle no equal sides. By finding the distances of each said we will tell whether a given triangle is equilateral, isosceles or scalene.

Complete step-by-step answer:
Given vertices are (-2, 0), (2, 3) and (1,-3).
We know that the distance between two points \[({x_1},{y_1})\] and \[({x_2},{y_2})\] is \[ = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_2} - {y_1})}^2}} \] .
Assume the diagram (for better understanding).

Now we find the distances of AB, AB and BC.
Now, the distance \[AB\] is given by,
The vertices of A and B are \[({x_1},{y_1}) = ( - 2,0)\] and \[({x_2},{y_2}) = (2,3)\]
Substituting we get,
 \[AB = \sqrt {{{( - 2 - 2)}^2} + {{(0 - 3)}^2}} \]
 \[ = \sqrt {{{( - 4)}^2} + {{( - 3)}^2}} \]
 \[ = \sqrt {16 + 9} \]
 \[ = \sqrt {25} \]
 \[ = 5\] units
Now, the distance of \[BC\] is given by,
The vertices of B and C are \[({x_1},{y_1}) = (2,3)\] and \[({x_2},{y_2}) = (1, - 3)\] .
 \[BC = \sqrt {{{(2 - 1)}^2} + {{(3 - ( - 3))}^2}} \]
 \[ = \sqrt {{{(1)}^2} + {{(6)}^2}} \]
 \[ = \sqrt {1 + 36} \]
 \[ = \sqrt {37} \] units.
Now, the distance of \[AC\] is given by,
The vertices of A and C are \[({x_1},{y_1}) = ( - 2,0)\] and \[({x_2},{y_2}) = (1, - 3)\] .
 \[AC = \sqrt {{{( - 2 - 1)}^2} + {{(0 - ( - 3))}^2}} \]
 \[ = \sqrt {{{( - 3)}^2} + {{(3)}^2}} \]
 \[ = \sqrt {9 + 9} \]
 \[ = \sqrt {18} \] units.
As we can see that the distance of the sides AB, BC, and AC are different.
Hence the given triangle is scalene.
So, the correct answer is “the given triangle is scalene”.

Note: Also we know that in equilateral triangles all the angels are the same. In the isosceles triangle only two angles are equal. In scalene all three angels are different. We can differentiate triangles based on angels also. The distance formula is the same for any two points. Do not get confused with the distance formula. The distance formula in above is the same as the \[ = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \] . Negative Square is positive.