Consider points 𝐴 one, one; 𝐵 negative one, one; and 𝐶 negative one, three in the orthonormal coordinate plane 𝑂; 𝐼, 𝐽. What type of coordinate plane is 𝑂; 𝐴, 𝐵? What are the coordinates of point 𝐶 in the coordinate plane 𝑂; 𝐴, 𝐵.
The first part of this question asked us to determine what type of coordinate plane 𝑂; 𝐴, 𝐵 is. The first letter, in this case 𝑂, corresponds to the origin. So 𝑂 has coordinates zero, zero. Line 𝑂𝐴 is the 𝑥-axis with 𝑂𝐴 as its unit length. And line 𝑂𝐵 is the 𝑦-axis with 𝑂𝐵 as its unit length. To identify the type of coordinate plane, we need to determine two things: firstly, where the lines 𝑂𝐴 and 𝑂𝐵 are perpendicular and, secondly, whether the length of 𝑂𝐴 is equal to the length of 𝑂𝐵.
We notice from the figure that line segments 𝑂𝐴 and 𝑂𝐵 are both diagonals of a square on the grid. We know that the diagonal of a square is an axis of symmetry of the square. This means that the measure of angle 𝐴𝑂𝐼 is equal to the measure of angle 𝐴𝑂𝐽. And both of these are equal to 45 degrees. In the same way, the measure of angle 𝐵𝑂𝐽 is 45 degrees. And since 45 plus 45 is equal to 90, the measure of angle 𝐴𝑂𝐵 is 90 degrees. And we can conclude that lines 𝑂𝐴 and 𝑂𝐵 are perpendicular.
We also note from the figure that as the diagonals of a square are of equal lengths and line segments 𝑂𝐴 and 𝑂𝐵 are diagonals of two congruent squares, we have 𝑂𝐴 is equal to 𝑂𝐵. Since the answer to both of our questions is yes, we can conclude that the plane 𝑂; 𝐴, 𝐵 is an orthonormal coordinate plane since its axes are perpendicular and they have the same unit length.
The second part of this question asks us to find the coordinates of point 𝐶 in this coordinate plane. Note that this is not the same as the coordinate of negative one, three as given in the question since these are the coordinates of point 𝐶 in the plane 𝑂; 𝐼, 𝐽. We can do this by firstly drawing two lines that are parallel to the 𝑥- and 𝑦-axes passing through point 𝐶. By the definition of the coordinate plane 𝐴; 𝑂, 𝐵, the length of line segment 𝑂𝐴 is one unit in the 𝑥-direction. And hence, the coordinates of point 𝐴 are one, zero. In the same way, the line segment 𝑂𝐵 is one unit in the 𝑦-direction. And 𝐵 has coordinates zero, one.
Since the line parallel to the 𝑦-axis passing through 𝐶 intersects the 𝑥-axis at 𝐴, we have an 𝑥-coordinate of one. The line parallel to the 𝑥-axis passing through 𝐶 intersects the 𝑦-axis at a point that is a distance from the origin twice the length of 𝑂𝐵. Since this is on the positive side of the 𝑦-axis, i.e, the same side as 𝐵, this corresponds to a 𝑦-coordinate of two. The coordinates of point 𝐶 in the coordinate plane 𝑂; 𝐴, 𝐵 are one, two.