Question Video: Properties of Oblique, Orthogonal, and Orthonormal Coordinate Planes Mathematics

Video Transcript

Consider the points 𝐴 zero, zero; 𝐵 one, zero; 𝐶 one, one; and 𝐷 zero, one in a coordinate plane. There are three parts to this question. Part one, if the coordinate plane is oblique, what is the shape of quadrilateral 𝐴𝐵𝐶𝐷? Is it (A) a square, (B) a kite, (C) a trapezoid, (D) a rectangle, or (E) a parallelogram?

In this question, we are asked to find the shape of quadrilateral 𝐴𝐵𝐶𝐷 given that our coordinate plane is oblique. We begin by recalling that when a coordinate plane is oblique, the 𝑥- and 𝑦-axes are not perpendicular. Marking on the points 𝐴, 𝐵, 𝐶, and 𝐷, a quadrilateral would look as shown. We see that line segment 𝐴𝐷 is parallel to line segment 𝐵𝐶, which is parallel to the 𝑦-axis. In the same way, line segment 𝐴𝐵 is parallel to line segment 𝐷𝐶, which is parallel to the 𝑥-axis.

The quadrilateral has two pairs of parallel sides which do not meet at 90 degrees. This means that if the coordinate plane is oblique, the shape of quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram. The correct answer is option (E).

We will now clear some space and consider the second part of this question.

In the second part to this question, we have the same four points 𝐴, 𝐵, 𝐶, 𝐷. This time, we’re asked if the coordinate plane is orthogonal, what is the shape of quadrilateral 𝐴𝐵𝐶𝐷? Option (A) a square, option (B) a kite, option (C) a trapezoid, or option (D) a rectangle.

The difference in this part of the question is that the coordinate plane is orthogonal and not oblique. We recall that this means that the 𝑥- and 𝑦-axes are perpendicular. Once again, we can mark on the points 𝐴, 𝐵, 𝐶, and 𝐷 as shown. Once again, line segments 𝐴𝐷 and 𝐵𝐶 are parallel to the 𝑦-axis, and line segments 𝐴𝐵 and 𝐷𝐶 are parallel to the 𝑥-axis. This means that the shape is once again a parallelogram. However, it is a special type of parallelogram, since line segments 𝐴𝐵 and 𝐴𝐷 are perpendicular.

A parallelogram with right angles is a rectangle. And we can therefore conclude that if the coordinate plane is orthogonal, the quadrilateral 𝐴𝐵𝐶𝐷 is a rectangle. The correct answer is option (D).

Let’s now consider the third and final part of this question.

We once again have the same four points. This time, we’re asked if the coordinate plane is orthonormal, what is the shape of quadrilateral 𝐴𝐵𝐶𝐷? This time, we have three options: (A) a square, (B) a kite, and (C) a trapezoid.

In an orthonormal coordinate plane, the 𝑥- and 𝑦-axes are perpendicular and the unit lengths are equal. This means that in the quadrilateral 𝐴𝐵𝐶𝐷, we have the same properties as the second part of this question. Line segments 𝐴𝐷 and 𝐵𝐶 are parallel to the 𝑦-axis, line segments 𝐴𝐵 and 𝐷𝐶 are parallel to the 𝑥-axis, and line segments 𝐴𝐵 and 𝐴𝐷 are perpendicular. We also have the additional property that the length of 𝐴𝐵 is equal to the length of 𝐴𝐷. As such, we can conclude that if the coordinate plane is orthonormal, the shape of quadrilateral 𝐴𝐵𝐶𝐷 is a square. The correct answer is option (A).

In summary, we found that for the points 𝐴 zero, zero; 𝐵 one, zero; 𝐶 one, one; and 𝐷 zero, one that if the coordinate plane is oblique, the shape of our quadrilateral 𝐴𝐵𝐶𝐷 is a parallelogram. If the coordinate plane is orthogonal, the quadrilateral is a rectangle. And if the coordinate plane is orthonormal, the shape of the quadrilateral is a square.