Which point is negative five, negative six on the coordinate plane 𝑂; 𝐼, 𝐽?
We begin by noting that the six points 𝐴 through 𝐹 lie in an orthonormal coordinate plane. We know this since the 𝑥- and 𝑦-axes are perpendicular and the unit lengths 𝑂𝐼 and 𝑂𝐽 in the 𝑥- and 𝑦-direction, respectively, are equal. We can see from the figure that one square represents one unit. As we are trying to identify the point with coordinates negative five, negative six, we need to move five units in the negative 𝑥-direction and six units in the negative 𝑦-direction from the origin 𝑂. Since both the 𝑥- and 𝑦-coordinates are negative, our point must lie in the third quadrant. This suggests that the correct answer must be point 𝐹.
We can check this by marking on negative five on the 𝑥-axis and negative six on the 𝑦-axis. We can then draw a line parallel to the 𝑦-axis that intersects the 𝑥-axis at negative five and a line parallel to the 𝑥-axis that intersects the 𝑦-axis at negative six. As these lines intersect at point 𝐹, we can conclude that the point 𝐹 has coordinates negative five, negative six on the coordinate plane 𝑂; 𝐼, 𝐽.