The distances satisfy the triangle inequality for each combination of sides.
XY forms a triangle.
YZ forms a triangle.
XZ forms a triangle.
How to examine the points
Let's determine whether the points X(1, 2) Y(4, 6) and Z(6, 6) can be the vertices of a triangle.
The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by the distance formula:
d = √{(x₂ - x₁)² + (y₂ - y₁)²}
Distance XY
=√((4 - 1)² + (6 - 2)²} = 5]
Distance YZ
=√{(6 - 4)² + (6 - 6)²} = 2
Distance XZ
= √{(6 - 1)² + (6 - 2)²} = √41
Now, to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Checking for the triangle inequality
1. 5 + 2 = 7 > XZ
2. 5 + √41 greater than YZ
3. √41 + 2 greater than XY
All three inequalities hold true, so the points X, Y, and Z can indeed be the vertices of a triangle.