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Practice
The Coordinate Plane
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The Coordinate Plane: Difficult Problems with Solutions
By Catalin David
Problem 1
If point A with the coordinates (2, a) is situated on the x-axis, then
a =
1
-1
0
2
.
Solution:
If point A is on the x-axis, the distance from A to the x-axis is 0, so a = 0.
Problem 2
If point B with the coordinates (a, 3) is situated on the y-axis, then a =
3
-1
1
0
.
Solution:
If point B is on the y-axis, the distance from B to the y-axis is 0, so a = 0.
Problem 3
What are the coordinates of a point found in quadrant II at 4 units from the y-axis and at 2 units from the x-axis?
(-2, 4)
(4, -2)
(-4, 2)
(-4, -2)
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).
Problem 4
What are the coordinates of a point found in quadrant III at 4 units from the x-axis and at 3 units from the y-axis?
(3, 4)
(-3 , -4)
(-3 , 4)
(3, -4)
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).
Problem 5
What are the coordinates of a point found in quadrant I at 3 units from the x-axis and at 2 units from the y-axis?
(3, 2)
(2, 3)
(-2, 3)
(2, -3)
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).
Problem 6
What are the coordinates of a point found in quadrant IV at 1 unit from the y-axis and at 4 units from the x-axis?
(1, 4)
(-1, 4)
(4, -1)
(1, -4)
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).
Problem 7
What is the distance between points A(1, 3) and B(4, 3)?
1
4
3
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.
Problem 8
What is the length of AB if A has the coordinates (-3, 2) and B has the coordinates (2, 2)?
-5
-1
5
1
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.
Problem 9
What is the length of AB if A has the coordinates (4, 2) and B has the coordinates (4, 6)?
2
6
4
-4
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.
Problem 10
What are the coordinates of the symmetric of point A with respect to point B?
(6, 3)
(8, 3)
(3, 8)
(-1, 3)
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).
Problem 11
What is the length of AB if A has the coordinates (-2, -5) and B has the coordinates (-2, 3)?
2
8
-3
7
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.
Problem 12
What is the length of AB if A has the coordinates (-4, 5) and B has the coordinates (-4, -1)?
5
1
6
4
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.
Problem 13
What are the coordinates of point B?
(3, 7)
(-1, 3)
(7, 3)
(7, 7)
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).
Problem 14
What are the coordinates of point B?
(3, 1)
(-1, 3)
(3, -1)
(3, 6)
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).
Problem 15
What are the coordinates of point C?
(2, 3)
(3, 2)
(1, 3)
(4, 2)
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).
Problem 16
What are the coordinates of point B?
(3, 2)
(2, -8)
(5, 2)
(5, -8)
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).
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