Video Transcript
𝐴𝐵𝐶𝐷 is a parallelogram, and points 𝐼, 𝐽, 𝐾, and 𝐿 are midpoints of the line segments 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴, respectively. Find the coordinates of point 𝐶 in each of the following coordinate plane configurations: the coordinate plane 𝐴; 𝐵, 𝐷, the coordinate plane 𝐶; 𝐷, 𝐵, the coordinate plane 𝐿; 𝐽, 𝐷, and the coordinate plane 𝐴; 𝐼, 𝐿.
Since 𝐴𝐵𝐶𝐷 is a parallelogram, we’re dealing with an oblique coordinate system. And as such, all four coordinate planes we’re asked to consider are oblique. We know this since the 𝑥- and 𝑦-axes of each plane are not perpendicular. In the first part of the question, we’re asked to consider the coordinate plane 𝐴; 𝐵, 𝐷. When the coordinate plane is written using this notation, the first letter corresponds to the origin. This means that point 𝐴 has coordinates zero, zero. The second letter means that 𝐴𝐵 is the 𝑥-axis with line segments 𝐴𝐵 its unit length. In this coordinate plane, point 𝐵 has coordinates one, zero. In the same way, line 𝐴𝐷 is the 𝑦-axis with the line segment 𝐴𝐷 its unit length. This means that 𝐷 has coordinates zero, one.
Since the angle 𝐷𝐴𝐵 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 𝐶. To find the 𝑥-coordinate, we look for a line parallel to the 𝑦-axis that passes through point 𝐶. This is the line 𝐵𝐶. And as it intersects the 𝑥-axis at point 𝐵, we know that the 𝑥-coordinate of 𝐶 is one. We can then repeat this process to find the 𝑦-coordinate of point 𝐶. We look for a line parallel to the 𝑥-axis that passes through point 𝐶. This is the line 𝐷𝐶. And it intersects the 𝑦-axis at point 𝐷. We can therefore conclude that point 𝐶 has coordinates one, one. If the coordinate plane is 𝐴; 𝐵, 𝐷, then point 𝐶 has coordinates one, one.
In the second part of this question, we need to consider the coordinate plane 𝐶; 𝐷, 𝐵. This time point 𝐶 is the origin. And as such, we don’t require any further information to conclude that in the coordinate plane 𝐶; 𝐷, 𝐵, point 𝐶 has coordinates zero, zero.
Next, we need to consider the coordinate plane 𝐿; 𝐽, 𝐷. This time the origin is at point 𝐿. The line 𝐿𝐽 is the 𝑥-axis with line segment 𝐿𝐽 its unit length. This means that 𝐽 has coordinates one, zero. 𝐿𝐷 is the 𝑦-axis with line segment 𝐿𝐷 its unit length. And 𝐷 therefore has coordinates zero, one. Repeating the process from the first part of this question, we see that line 𝐽𝐶 is parallel to the 𝑦-axis and intersects the 𝑥-axis at point 𝐽. Likewise, line 𝐷𝐶 is parallel to the 𝑥-axis, and this intersects the 𝑦-axis at point 𝐷. Point 𝐶 is therefore one unit in the 𝑥-direction and one unit in the 𝑦-direction. And we can conclude that in the oblique coordinate plane 𝐿; 𝐽, 𝐷, point 𝐶 has coordinates one, one.
Let’s now consider the final part of this question. This time we have the coordinate plane 𝐴; 𝐼, 𝐿. As with the first part of this question, the origin lies at point 𝐴. So 𝐴 has coordinates zero, zero. Line 𝐴𝐼 is the 𝑥-axis with line segment 𝐴𝐼 its unit length. And 𝐼 therefore has coordinates one, zero. Point 𝐿 has coordinates zero, one as 𝐴𝐿 is the 𝑦-axis and the line segment 𝐴𝐿 is its unit length. Since points 𝐼 and 𝐿 are the midpoints of the line segments 𝐴𝐵 and 𝐷𝐴, respectively, we know that in this oblique coordinate plane, point 𝐵 has coordinates two, zero and point 𝐷 has coordinates zero, two. We can then draw lines parallel to the 𝑥- and 𝑦-axes that pass through point 𝐶. Once again, as these lines intersect both the 𝑥- and 𝑦-axis at two, we can conclude that point 𝐶 has coordinates two, two.
We now have answers to all four parts of this question. Point 𝐶 has coordinates one, one; zero, zero; one, one; and two, two in the four given oblique coordinate planes.
