# Question Video: Identifying the Coordinates of Points in an Oblique Coordinate System Mathematics

π΄π΅πΆπ· is a parallelogram, and points πΌ, π½, πΎ, and πΏ are midpoints of the line segments π΄π΅, π΅πΆ, πΆπ·, and π·π΄, respectively. Find the coordinates of point πΆ in each of the following coordinate plane configurations: The coordinate plane (π΄; π΅, π·), The coordinate plane (πΆ; π·, π΅), The coordinate plane (πΏ; π½, π·), The coordinate plane (π΄; πΌ, πΏ)

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### Video Transcript

π΄π΅πΆπ· is a parallelogram, and points πΌ, π½, πΎ, and πΏ are midpoints of the line segments π΄π΅, π΅πΆ, πΆπ·, and π·π΄, respectively. Find the coordinates of point πΆ in each of the following coordinate plane configurations: the coordinate plane π΄; π΅, π·, the coordinate plane πΆ; π·, π΅, the coordinate plane πΏ; π½, π·, and the coordinate plane π΄; πΌ, πΏ.

Since π΄π΅πΆπ· is a parallelogram, weβre dealing with an oblique coordinate system. And as such, all four coordinate planes weβre 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, weβre 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 πΆ. To find the π₯-coordinate, we look for a line parallel to the π¦-axis that passes through point πΆ. This is the line π΅πΆ. And as it intersects the π₯-axis at point π΅, we know that the π₯-coordinate of πΆ is one. We can then repeat this process to find the π¦-coordinate of point πΆ. We look for a line parallel to the π₯-axis that passes through point πΆ. This is the line π·πΆ. And it intersects the π¦-axis at point π·. We can therefore conclude that point πΆ has coordinates one, one. If the coordinate plane is π΄; π΅, π·, then point πΆ has coordinates one, one.

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

Next, we need to consider the coordinate plane πΏ; π½, π·. This time the origin is at point πΏ. The line πΏπ½ is the π₯-axis with line segment πΏπ½ its unit length. This means that π½ has coordinates one, zero. πΏπ· is the π¦-axis with line segment πΏπ· its unit length. And π· therefore has coordinates zero, one. Repeating the process from the first part of this question, we see that line π½πΆ is parallel to the π¦-axis and intersects the π₯-axis at point π½. Likewise, line π·πΆ is parallel to the π₯-axis, and this intersects the π¦-axis at point π·. Point πΆ 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 πΏ; π½, π·, point πΆ has coordinates one, one.

Letβs now consider the final part of this question. This time we have the coordinate plane π΄; πΌ, πΏ. As with the first part of this question, the origin lies at point π΄. So π΄ has coordinates zero, zero. Line π΄πΌ is the π₯-axis with line segment π΄πΌ its unit length. And πΌ therefore has coordinates one, zero. Point πΏ has coordinates zero, one as π΄πΏ is the π¦-axis and the line segment π΄πΏ is its unit length. Since points πΌ and πΏ 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 πΆ. Once again, as these lines intersect both the π₯- and π¦-axis at two, we can conclude that point πΆ has coordinates two, two.

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