To specify the position of a figure in the coordinate plane, we need two variables for each vertex. With special positioning and using special properties, we can make sure that the figure we get has the desired properties. Below are examples.

General Triangle

Sometimes it is useful to place a vertex of a triangle at the origin and another vertex on the $x -$ axis.

Isosceles Triangle

Placing the base of an isosceles triangle on the $x -$ axis with the midpoint in the origin and the third vertex on the $y -$ axis guarantees that the triangle we get is isosceles.

Quadrilaterals with Parallel Sides

To get a general trapezoid, we can place one side along the $x -$ axis with one vertex at the origin. To guarantee that the quadrilateral has two parallel sides, we can choose the $y -$ coordinates of the other two vertices to be the same.

If we place the midpoints of the bases of a trapezoid on the $y -$ axis, then we get a trapezoid with congruent legs, so we get an isosceles trapezoid.

To get a parallelogram, we need to make sure it has two pairs of parallel sides. This can be done by placing three vertices like we did with the trapezoid above and expressing the coordinates of the fourth vertex in terms of the coordinates of the other three.

A rectangle is a parallelogram where the angles are right angles. We can make sure we get a shape like this if two sides of a parallelogram are placed on the coordinate axes.

A square is a rectangle with congruent sides. To get a shape like this, we can choose the width and length of a rectangle to be the same.

Quadrilaterals with Perpendicular Diagonals

Kites are quadrilaterals with perpendicular diagonals. Moreover, one diagonal bisects the other. We can get a shape like this if we place the diagonals on the coordinate axes and we make sure that on the $y -$ axis the origin is the midpoint of the diagonal.

The last quadrilateral we look at is the rhombus. In a rhombus the diagonals are perpendicular and bisect each other. We can get a shape like this if we place the diagonals on the coordinate axes and make sure that the origin is the midpoint of both.

Conclusion

All of the above graphs show examples of how we can use variable coordinates to generalize figures in the coordinate plane.

Exercise