### Video Transcript

π΄π΅πΆπ· is a parallelogram, and points πΌ, π½, πΎ, and πΏ are midpoints of the line segments π΄π΅, π΅πΆ, πΆπ·, and π·π΄, respectively. Find the coordinates of point πΆ in each of the following coordinate plane configurations: the coordinate plane π΄; π΅, π·, the coordinate plane πΆ; π·, π΅, the coordinate plane πΏ; π½, π·, and the coordinate plane π΄; πΌ, πΏ.

Since π΄π΅πΆπ· is a parallelogram, weβre dealing with an oblique coordinate system. And as such, all four coordinate planes weβre asked to consider are oblique. We know this since the π₯- and π¦-axes of each plane are not perpendicular. In the first part of the question, weβre asked to consider the coordinate plane π΄; π΅, π·. When the coordinate plane is written using this notation, the first letter corresponds to the origin. This means that point π΄ has coordinates zero, zero. The second letter means that π΄π΅ is the π₯-axis with line segments π΄π΅ its unit length. In this coordinate plane, point π΅ has coordinates one, zero. In the same way, line π΄π· is the π¦-axis with the line segment π΄π· its unit length. This means that π· has coordinates zero, one.

Since the angle π·π΄π΅ is not equal to 90 degrees, this confirms we are dealing with an oblique coordinate plane. In each part of this question, we need to find the coordinates of point πΆ. To find the π₯-coordinate, we look for a line parallel to the π¦-axis that passes through point πΆ. This is the line π΅πΆ. And as it intersects the π₯-axis at point π΅, we know that the π₯-coordinate of πΆ is one. We can then repeat this process to find the π¦-coordinate of point πΆ. We look for a line parallel to the π₯-axis that passes through point πΆ. This is the line π·πΆ. And it intersects the π¦-axis at point π·. We can therefore conclude that point πΆ has coordinates one, one. If the coordinate plane is π΄; π΅, π·, then point πΆ has coordinates one, one.

In the second part of this question, we need to consider the coordinate plane πΆ; π·, π΅. This time point πΆ is the origin. And as such, we donβt require any further information to conclude that in the coordinate plane πΆ; π·, π΅, point πΆ has coordinates zero, zero.

Next, we need to consider the coordinate plane πΏ; π½, π·. This time the origin is at point πΏ. The line πΏπ½ is the π₯-axis with line segment πΏπ½ its unit length. This means that π½ has coordinates one, zero. πΏπ· is the π¦-axis with line segment πΏπ· its unit length. And π· therefore has coordinates zero, one. Repeating the process from the first part of this question, we see that line π½πΆ is parallel to the π¦-axis and intersects the π₯-axis at point π½. Likewise, line π·πΆ is parallel to the π₯-axis, and this intersects the π¦-axis at point π·. Point πΆ is therefore one unit in the π₯-direction and one unit in the π¦-direction. And we can conclude that in the oblique coordinate plane πΏ; π½, π·, point πΆ has coordinates one, one.

Letβs now consider the final part of this question. This time we have the coordinate plane π΄; πΌ, πΏ. As with the first part of this question, the origin lies at point π΄. So π΄ has coordinates zero, zero. Line π΄πΌ is the π₯-axis with line segment π΄πΌ its unit length. And πΌ therefore has coordinates one, zero. Point πΏ has coordinates zero, one as π΄πΏ is the π¦-axis and the line segment π΄πΏ is its unit length. Since points πΌ and πΏ are the midpoints of the line segments π΄π΅ and π·π΄, respectively, we know that in this oblique coordinate plane, point π΅ has coordinates two, zero and point π· has coordinates zero, two. We can then draw lines parallel to the π₯- and π¦-axes that pass through point πΆ. Once again, as these lines intersect both the π₯- and π¦-axis at two, we can conclude that point πΆ has coordinates two, two.

We now have answers to all four parts of this question. Point πΆ has coordinates one, one; zero, zero; one, one; and two, two in the four given oblique coordinate planes.