### Video Transcript

Consider the points π΄ zero, zero; π΅ one, zero; πΆ one, one; and π· 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 π΄π΅πΆπ·? 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 π΄π΅πΆπ· given that our coordinate plane is oblique. We begin by recalling that when a coordinate plane is oblique, the π₯- and π¦-axes are not perpendicular. Marking on the points π΄, π΅, πΆ, and π·, a quadrilateral would look as shown. We see that line segment π΄π· is parallel to line segment π΅πΆ, which is parallel to the π¦-axis. In the same way, line segment π΄π΅ is parallel to line segment π·πΆ, which is parallel to the π₯-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 π΄π΅πΆπ· 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 π΄, π΅, πΆ, π·. This time, weβre asked if the coordinate plane is orthogonal, what is the shape of quadrilateral π΄π΅πΆπ·? 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 π₯- and π¦-axes are perpendicular. Once again, we can mark on the points π΄, π΅, πΆ, and π· as shown. Once again, line segments π΄π· and π΅πΆ are parallel to the π¦-axis, and line segments π΄π΅ and π·πΆ are parallel to the π₯-axis. This means that the shape is once again a parallelogram. However, it is a special type of parallelogram, since line segments π΄π΅ and π΄π· are perpendicular.

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

Letβs now consider the third and final part of this question.

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

In an orthonormal coordinate plane, the π₯- and π¦-axes are perpendicular and the unit lengths are equal. This means that in the quadrilateral π΄π΅πΆπ·, we have the same properties as the second part of this question. Line segments π΄π· and π΅πΆ are parallel to the π¦-axis, line segments π΄π΅ and π·πΆ are parallel to the π₯-axis, and line segments π΄π΅ and π΄π· are perpendicular. We also have the additional property that the length of π΄π΅ is equal to the length of π΄π·. As such, we can conclude that if the coordinate plane is orthonormal, the shape of quadrilateral π΄π΅πΆπ· is a square. The correct answer is option (A).

In summary, we found that for the points π΄ zero, zero; π΅ one, zero; πΆ one, one; and π· zero, one that if the coordinate plane is oblique, the shape of our quadrilateral π΄π΅πΆπ· 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.