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The Coordinate Plane: Problems with Solutions
By Catalin David
Problem 1
The x-axis is
horizontal
vertical
.
Solution:
Horizontal
Problem 2
The y-axis is
horizontal
vertical
.
Solution:
Vertical
Problem 3
How many coordinates does any point in a plane have?
1
2
3
4
Solution:
2
Problem 4
The first coordinate of the point is called
x
y
.
Solution:
The x coordinate, which is always on the x-axis.
Problem 5
The second coordinate of the point is
x
y
.
Solution:
The y coordinate, which is always on the y-axis.
Problem 6
The x coordinate of a point in a plane represents
the distance from the point to the x-axis
the distance from the point to the y-axis
.
Solution:
The distance from the point to the y-axis.
Problem 7
The y coordinate of a point in a plane represents
the distance from the point to the x-axis
the distance from the point to the y-axis
.
Solution:
The distance from the point to the x-axis.
Problem 8
How many points on the y-axis are at a distance of 3 units from the x-axis?
1
2
Solution:
If the points are on the y-axis, their first coordinate is 0. If the points are at a distance of 3 units from the x-axis, their second coordinate can be 3 or -3. There are two points which fulfill both conditions: B(0, 3) and B'(0, -3).
Problem 9
How many points on the x-axis are at a distance of 2 units from the y-axis?
1
2
Solution:
If the points are on the x-axis, their second coordinate is 0. If the points are at a distance of 2 units from the y-axis, their first coordinate can be 2 or -2. There are two points which fulfill both conditions: A(2, 0) and A'(-2, 0).
Problem 10
How many quadrants are formed by the x- and y-axes in a plane?
2
4
1
6
Solution:
4
Problem 11
The coordinates of the point where the x- and y-axes meet are
(0, 0)
(0, 1)
(1, 0)
(-1, 1)
.
Solution:
(0, 0)
Problem 12
What are the signs of the two coordinates of point A?
(-, +)
(-, -)
(+, +)
(+, -)
Solution:
Point A is found in quadrant I,
so both its coordinates are positive.
Problem 13
What are the signs of the two coordinates of point B?
(-, -)
(+, +)
(+, -)
(-, +)
Solution:
Point B is found in quadrant IV,
so its x-coordinate is positive and
its y-coordinate is negative.
Problem 14
What are the signs of the two coordinates of point C?
(+, -)
(-, -)
(+, +)
(-, +)
Solution:
Point C is found in quadrant II,
so its x-coordinate is negative and
its y-coordinate is positive.
Problem 15
What are the signs of the two coordinates of point D?
(+, -)
(-, -)
(+, +)
(-, +)
Solution:
Point D is found in quadrant III,
so both its coordinates are negative.
Problem 16
Which quadrant contains point A (-1, -1)?
II
IV
III
I
Solution:
Since both its coordinates are negative, point A is in quadrant III.
Problem 17
Which quadrant contains point B (-2, 1)?
I
II
III
IV
Solution:
Since its first coordinate is negative and the second one is positive, point B is in quadrant II.
Problem 18
What are the coordinates of point A?
(2, 1)
(1, 2)
(2, 2)
(1, 1)
Solution:
The coordinates of point A are (2, 1), since it is found in quadrant I at a distance of 2 units of length from the y-axis and 1 unit from the x-axis.
Problem 19
What are the coordinates of point B?
(2, 3)
(3, -2)
(-2, 3)
(-2, -3)
Solution:
The coordinates of point B are (-2, 3), since it is found in quadrant II at a distance of 2 units of length from the y-axis and 3 units from the x-axis.
Problem 20
What are the coordinates of point C?
(-4, 3)
(3, -4)
(-3, 4)
(-3, -4)
Solution:
The coordinates of point C are (3, -4), since it is found in quadrant III at a distance of 3 units of length from the y-axis and 4 units from the x-axis.
Problem 21
What are the coordinates of point D?
(1, -1)
(1, 1)
(-1, 1)
(-1, -1)
Solution:
The coordinates of point D are (-1, -1), since it is found in quadrant IV at a distance of 1 units of length from the Y-axis and 1 unit from the x-axis.
Problem 22
What are the coordinates of point A?
(0, 3)
(3, 3)
(-3, 0)
(3, 0)
Solution:
Point A is right on the x-axis, so its y-coordinate is 0. A is at a distance of 3 units to the right of the y-axis, so its x-coordinate is 3. Thus, the coordinates of point A are (3, 0).
Problem 23
What are the coordinates of point B?
(-2, 0)
(0, -2)
(0, 2)
(1, 1)
Solution:
Point B is right on the y-axis, so its x-coordinate is 0. B is at a distance of 2 units from the x-axis, so its y-coordinate is 2. Thus, the coordinates of point B are (0, -2).
Problem 24
Which point has the coordinates (-3, 4)?
A
B
C
D
Solution:
Answer: A
Problem 25
What is the length of BC?
2
3
6
4
Solution:
Since points B and C have the same y-coordinate, the length of BC is equal to the difference of the distances from the two points to the y-axis. The coordinates of B are (3, 2) and those of C are (6, 2). B is at a distance of 3 units from the y-axis and C is at a distance of 6 units from the y-axis. Thus, the length of BC is 6 - 3 = 3 units.
Problem 26
What is the length of BC?
3
2
4
5
Solution:
Since points B and C have the same x-coordinate, the length of BC is equal to the difference of the distances from the two points to the x-axis. The coordinates of B are (3, 2) and those of C are (3, 4). B is at a distance of 2 units from the x-axis and C is at a distance of 4 units from the x-axis. Thus, the length of BC is 4 - 2 = 2 units.
Problem 27
What is the length of AB?
3
2
5
1
Solution:
Since point A's coordinates are (4, 3), the distance from A to the x-axis is 3 units. Since point B's coordinates are (4, -2), the distance from B to the x-axis is 2 units. The length of AB is 3 + 2 = 5 units.
Problem 28
What is the length of AB?
3
9
6
8
Solution:
Since point A's coordinates are (-3, 1), the distance from A to the y-axis is 3 units. Since point B's coordinates are (6, 1), the distance from B to the y-axis is 6 units. The length of AB is 3 + 6 = 9 units.
Problem 29
What is the perimeter of rectangle ABCD?
6
10
12
15
Solution:
The perimeter of a rectangle is 2L + 2l (= 2AB + 2BC).
Since point A's coordinates are (2, 3), the distance from A to the y-axis is 2 units. Since point B's coordinates are (6, 3), the distance from B to the y-axis is 6 units. Thus, AB = 6 - 2 = 4 units.
Since point C's coordinates are (6, 1), the distance from C to the x-axis is 1 unit and the distance from B to the x-axis is 3. Thus, BC = 3 - 1 = 2 units.
P = $2 \cdot 4 + 2 \cdot 2 = 8 + 4 = 12$
Problem 30
What are the coordinates of the symmetric of A with respect to the x-axis?
(-2, 3)
(2, -3)
(-2, -3)
Solution:
The symmetric of point A with respect to the x-axis is at the same distance from the x-axis as A. Thus, its second coordinate is -3, the first coordinate remaining the same. The coordinates of the point are (2, -3).
Problem 31
What are the coordinates of the symmetric of point A with respect to the y-axis?
(2, -3)
(-2, -3)
(-2, 3)
Solution:
The symmetric of point A with respect to the y-axis is at the same distance from the y-axis as A. Thus, its first coordinate is -2, the second coordinate remaining the same. The coordinates of the point are (-2, 3).
Problem 32
If point A moved 2 units higher, what would its coordinates be?
(5, 2)
(4, 3)
(2, 5)
(2, 1)
Solution:
If point A moved 2 units higher, its y-coordinate would be 3 + 2 = 5. The x-coordinate would remain the same. The new coordinates would be (2, 5).
Problem 33
If point B moved 2 units higher, what would its coordinates be?
(3, -6)
(5, -4)
(3, -2)
(3, -4)
Solution:
If point B moved 2 units higher, its y-coordinate would be -4 + 2 = -2. The x-coordinate would remain the same. The new coordinates would be (3, -2).
Problem 34
If point A moved 1 unit lower, what would its coordinates be?
(1, 3)
(2, 2)
(1, 2)
(4, 2)
Solution:
If point A moved 1 unit lower, its y-coordinate would be 3 - 1 = 2. The x-coordinate would remain the same. The new coordinates would be (2, 2).
Problem 35
If point B moved 2 units lower, what would its coordinates be?
(1, -4)
(1, -6)
(3, -6)
Solution:
If point B moved 2 units lower, its y-coordinate would be -4 - 2 = -6. The x-coordinate would remain the same. The new coordinates would be (3, -6).
Problem 36
If point A moved 3 units to the left, what would its coordinates be?
(2, 0)
(-1, 0)
(-1, 3)
Solution:
If point A moved 3 units to the left, its x-coordinate would be 2 - 3 = -1. The y-coordinate would remain the same. The new coordinates would be (-1, 3).
Problem 37
If point B moved 2 units to the right, what would its coordinates be?
(3, -2)
(5, -4)
(5, -2)
(1, -4)
Solution:
If point B moved 2 units to the right, its x-coordinate would be 3 + 2 = 5. The y-coordinate would remain the same. The new coordinates would be (5, -4).
Easy
Normal
Difficult
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