The Coordinate Plane: Difficult Problems with Solutions

Solution:
If point A is on the x-axis, the distance from A to the x-axis is 0, so a = 0.

Solution:
If point B is on the y-axis, the distance from B to the y-axis is 0, so a = 0.

Solution:
Since the point is found in quadrant II, its first coordinate is negative and its second coordinate is positive. Since it is at a distance of 4 units from the y-axis, its first coordinate is -4. Since it is at a distance of 2 units from the x-axis, its second coordinate is 2. The coordinates of the point are (-4, 2).

Solution:
Since the point is found in quadrant III, both its coordinates are negative. Since it is at a distance of 4 units from the x-axis, its second coordinate is -4. Since it is at a distance of 3 units from the y-axis, its first coordinate is -3. The coordinates of the point are (-3, -4).

Solution:
Since the point is found in quadrant I, both its coordinates are positive. Since it is at a distance of 3 units from the x-axis, its second coordinate is 3. Since it is at a distance of 2 units from the y-axis, its first coordinate is 2. The coordinates of the point are (2, 3).

Solution:
Since the point is found in quadrant IV, its first coordinate is positive and its second coordinate is negative. Since it is at a distance of 1 unit from the y-axis, its first coordinate is 1. Since it is at a distance of 4 units from the x-axis, its second coordinate is -4. The coordinates of the point are (1, -4).

Solution:
Since the points have the same y-coordinate, AB is parallel to the x-axis. The distance between the 2 points is equal to the difference of the x-coordinates. The distance between point A and point B is 4 - 1 = 3 units.

Solution:
The length of AB is equal to the distance from A to B. Since both points have the same y-coordinate, AB is parallel to the x-axis. The length of AB is equal to the difference of the x-coordinates, so 2 - (-3) = 2 + 3 = 5.

Solution:
The length of AB is equal to the distance from A to B. Since both points have the same x-coordinate, AB is parallel to the y-axis. The length of AB is equal to the difference of the y-coordinates, so 6 - 2 = 4.

Solution:
The symmetric of point A with respect to point B is point A', which is found on the same line as A and B. B is halfway between A and A'. The distance from A' to the y-axis is 6 units greater, so its first coordinate is 8. The distance to the x-axis is the same, so the second coordinate of point A' is 3.
The coordinates of point A' are (8, 3).

Solution:
The length of AB is equal to the distance from A to B. Since both points have the same x-coordinate, AB is parallel to the y-axis. The length of AB is equal to the difference of the y-coordinates, so 3 - (-5) = 8.

Solution:
The length of AB is equal to the distance from A to B. Since both points have the same x-coordinate, AB is parallel to the y-axis. The length of AB is equal to the difference of the y-coordinates, so 5 - (-1) = 5 + 1 = 6.

Solution:
Since point A has the coordinates (3, 3) and AB is horizontal, point B will have the same y-coordinate. Since the length of AB is 4, the x-coordinate of point B is 3 + 4 = 7. Point B has the coordinates (7, 3).

Solution:
Since point A has the coordinates (3, 3) and AB is vertical, point B will have the same x-coordinate. Since the length of AB is 4, the y-coordinate of point B is 3 - 4 = -1. Point B has the coordinates (3, -1).

Solution:
Since AB is horizontal, the y-coordinate of point B is 1 and the x-coordinate is 1 + 2 = 3. Thus, B has the coordinates (3, 1). Since BC is vertical, the x-coordinate of point C is 3 and the y-coordinate is 1 + 1 = 2. Thus, point C has the coordinates (3, 2).

Solution:
Since all horizontal lines have a length of 1 unit, the x-coordinate of point B is 1 + 4 = 5.
Since all vertical lines have a length of 1 unit, the y-coordinate of point B is -3 + 5 = 2.
Point B has the coordinates (5, 2).