Worksheet - Coordinate Plane Basics

DIRECTIONS: Each question has five answer choices. Select the one best answer. Do not use a calculator.

Section A - Coordinate Plane Fundamentals - Questions 1 to 7

1. Point A is located at \((-3, 5)\) on the coordinate plane. In which quadrant does point A lie?

1. Quadrant I
2. Quadrant II
3. Quadrant III
4. Quadrant IV
5. On an axis

2. What are the coordinates of the origin?

1. \((1, 1)\)
2. \((0, 1)\)
3. \((1, 0)\)
4. \((0, 0)\)
5. \((-1, -1)\)

3. Point B has coordinates \((4, -7)\). What is the x-coordinate of point B?

1. -7
2. 4
3. -4
4. 7
5. 0

4. A point lies on the y-axis. Which of the following could be its coordinates?

1. \((3, 0)\)
2. \((0, -5)\)
3. \((2, 2)\)
4. \((-1, 4)\)
5. \((5, 5)\)

5. Point C is located at \((6, 0)\). Where does this point lie?

1. On the y-axis
2. On the x-axis
3. In Quadrant I
4. In Quadrant IV
5. At the origin

6. In which quadrant do both coordinates of any point have negative values?

1. Quadrant I
2. Quadrant II
3. Quadrant III
4. Quadrant IV
5. None; this occurs only at the origin

7. Point D has coordinates \((x, y)\) where \(x > 0\) and \(y < 0\).="" in="" which="" quadrant="" is="" point="" d="">

1. Quadrant I
2. Quadrant II
3. Quadrant III
4. Quadrant IV
5. Cannot be determined

Section B - Distance and Midpoint - Questions 8 to 14

8. What is the distance between points \((2, 3)\) and \((2, 8)\)?

1. 3
2. 5
3. 6
4. 10
5. 11

9. What is the distance between points \((-1, 4)\) and \((5, 4)\)?

1. 4
2. 5
3. 6
4. 7
5. 8

10. The midpoint of a line segment with endpoints \((0, 0)\) and \((6, 8)\) has coordinates

1. \((3, 4)\)
2. \((6, 8)\)
3. \((2, 3)\)
4. \((4, 6)\)
5. \((12, 16)\)

11. What is the distance between points \((1, 2)\) and \((4, 6)\)?

1. 3
2. 4
3. 5
4. 6
5. 7

12. The midpoint of a segment is \((3, 5)\). If one endpoint is \((1, 2)\), what is the other endpoint?

1. \((2, 3.5)\)
2. \((4, 7)\)
3. \((5, 8)\)
4. \((2, 7)\)
5. \((5, 3)\)

13. Point P is at \((-2, 3)\) and point Q is at \((4, 3)\). What is the distance from P to Q?

1. 2
2. 4
3. 6
4. 8
5. 10

14. What is the distance from the origin to the point \((5, 12)\)?

1. 7
2. 11
3. 13
4. 17
5. 119

Section C - Advanced Application - Questions 15 to 20

15. A rectangle has vertices at \((0, 0)\), \((0, 5)\), \((8, 5)\), and \((8, 0)\). What is the perimeter of the rectangle?

1. 13
2. 18
3. 26
4. 40
5. 80

16. Point A is at \((3, 7)\). If point A is reflected across the x-axis to create point B, what are the coordinates of point B?

1. \((-3, 7)\)
2. \((3, -7)\)
3. \((-3, -7)\)
4. \((7, 3)\)
5. \((7, -3)\)

17. Three vertices of a square are located at \((1, 1)\), \((1, 4)\), and \((4, 4)\). What are the coordinates of the fourth vertex?

1. \((4, 1)\)
2. \((1, 7)\)
3. \((4, 7)\)
4. \((7, 4)\)
5. \((3, 3)\)

18. Point M is the midpoint of segment PQ. If M is at \((2, 6)\) and P is at \((-1, 4)\), what is the distance from P to Q?

1. \(\sqrt{13}\)
2. \(2\sqrt{13}\)
3. 5
4. 10
5. 13

19. A point in Quadrant II is reflected across the y-axis. In which quadrant will the reflected point lie?

1. Quadrant I
2. Quadrant II
3. Quadrant III
4. Quadrant IV
5. On the x-axis

20. Points A, B, and C form a right triangle with the right angle at B. If A is at \((1, 1)\), B is at \((1, 5)\), and C is at \((7, 5)\), what is the area of triangle ABC?

1. 6
2. 10
3. 12
4. 20
5. 24

Answer Key

Quick Reference

1 B 2 D 3 B 4 B 5 B 6 C 7 D 8 B 9 C 10 A

11 C 12 C 13 C 14 C 15 C 16 B 17 A 18 B 19 A 20 C

Detailed Explanations

Question 1 - Correct Answer: B

The coordinates of point A are \((-3, 5)\).
The x-coordinate is -3, which is negative.
The y-coordinate is 5, which is positive.
Quadrant II contains all points with negative x-coordinates and positive y-coordinates.
Point A lies in Quadrant II.

Choice A is incorrect because Quadrant I has both coordinates positive, but the x-coordinate of A is negative.

Question 2 - Correct Answer: D

The origin is the point where the x-axis and y-axis intersect.
At this intersection, both coordinates are zero.
The coordinates of the origin are \((0, 0)\).

Choice A is incorrect because \((1, 1)\) is a point in Quadrant I, not at the intersection of the axes.

Question 3 - Correct Answer: B

The coordinates of point B are given as \((4, -7)\).
In an ordered pair \((x, y)\), the first value is the x-coordinate.
The x-coordinate of point B is 4.

Choice A is incorrect because -7 is the y-coordinate, not the x-coordinate.

Question 4 - Correct Answer: B

A point lies on the y-axis when its x-coordinate is 0.
Check each choice for an x-coordinate of 0.
Choice B, \((0, -5)\), has x-coordinate equal to 0.
This point lies on the y-axis.

Choice A is incorrect because \((3, 0)\) has x-coordinate 3, so it lies on the x-axis instead.

Question 5 - Correct Answer: B

Point C has coordinates \((6, 0)\).
The y-coordinate is 0.
Points with y-coordinate equal to 0 lie on the x-axis.
Point C lies on the x-axis.

Choice A is incorrect because points on the y-axis have x-coordinate 0, but point C has x-coordinate 6.

Question 6 - Correct Answer: C

Quadrant I: x positive, y positive.
Quadrant II: x negative, y positive.
Quadrant III: x negative, y negative.
Quadrant IV: x positive, y negative.
Both coordinates are negative in Quadrant III.

Choice B is incorrect because in Quadrant II the x-coordinate is negative but the y-coordinate is positive.

Question 7 - Correct Answer: D

Point D has coordinates \((x, y)\) with \(x > 0\) and \(y <>
The x-coordinate is positive.
The y-coordinate is negative.
Quadrant IV contains all points with positive x-coordinates and negative y-coordinates.
Point D is in Quadrant IV.

Choice A is incorrect because Quadrant I has both coordinates positive, but y is negative for point D.

Question 8 - Correct Answer: B

The two points are \((2, 3)\) and \((2, 8)\).
Both points have the same x-coordinate, so they lie on a vertical line.
The distance between them is the absolute difference of the y-coordinates.
\(|8 - 3| = 5\).
The distance is 5.

Choice A is incorrect because 3 is one of the y-coordinates, not the difference between them.

Question 9 - Correct Answer: C

The two points are \((-1, 4)\) and \((5, 4)\).
Both points have the same y-coordinate, so they lie on a horizontal line.
The distance between them is the absolute difference of the x-coordinates.
\(|5 - (-1)| = |5 + 1| = 6\).
The distance is 6.

Choice B is incorrect because 5 is one of the x-coordinates, not the horizontal distance between the points.

Question 10 - Correct Answer: A

The endpoints are \((0, 0)\) and \((6, 8)\).
The midpoint formula is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
\(\frac{0 + 6}{2} = \frac{6}{2} = 3\).
\(\frac{0 + 8}{2} = \frac{8}{2} = 4\).
The midpoint is \((3, 4)\).

Choice E is incorrect because \((12, 16)\) results from doubling the second endpoint instead of averaging the coordinates.

Question 11 - Correct Answer: C

The two points are \((1, 2)\) and \((4, 6)\).
Use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
\(d = \sqrt{(4 - 1)^2 + (6 - 2)^2}\).
\(d = \sqrt{3^2 + 4^2}\).
\(d = \sqrt{9 + 16}\).
\(d = \sqrt{25}\).
\(d = 5\).

Choice B is incorrect because 4 is the difference in y-coordinates alone, not the full distance.

Question 12 - Correct Answer: C

The midpoint is \((3, 5)\) and one endpoint is \((1, 2)\).
Let the other endpoint be \((x, y)\).
Using the midpoint formula: \(\frac{1 + x}{2} = 3\) and \(\frac{2 + y}{2} = 5\).
From the first equation: \(1 + x = 6\), so \(x = 5\).
From the second equation: \(2 + y = 10\), so \(y = 8\).
The other endpoint is \((5, 8)\).

Choice B is incorrect because \((4, 7)\) results from incorrectly adding 3 to each coordinate of the given endpoint instead of using the midpoint formula properly.

Question 13 - Correct Answer: C

Point P is at \((-2, 3)\) and point Q is at \((4, 3)\).
Both points have the same y-coordinate, so they lie on a horizontal line.
The distance is the absolute difference of the x-coordinates.
\(|4 - (-2)| = |4 + 2| = 6\).
The distance from P to Q is 6.

Choice B is incorrect because 4 is the x-coordinate of Q, not the distance between the two points.

Question 14 - Correct Answer: C

The origin is at \((0, 0)\) and the point is \((5, 12)\).
Use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
\(d = \sqrt{(5 - 0)^2 + (12 - 0)^2}\).
\(d = \sqrt{5^2 + 12^2}\).
\(d = \sqrt{25 + 144}\).
\(d = \sqrt{169}\).
\(d = 13\).

Choice E is incorrect because 119 results from adding 25 and 144 without taking the square root, and also computing the sum incorrectly.

Question 15 - Correct Answer: C

The vertices are \((0, 0)\), \((0, 5)\), \((8, 5)\), and \((8, 0)\).
The length is the distance from \((0, 0)\) to \((8, 0)\), which is 8.
The width is the distance from \((0, 0)\) to \((0, 5)\), which is 5.
The perimeter of a rectangle is \(2 \times (\text{length} + \text{width})\).
\(P = 2 \times (8 + 5) = 2 \times 13 = 26\).

Choice D is incorrect because 40 results from multiplying the length by the width to find area, then not applying the perimeter formula correctly.

Question 16 - Correct Answer: B

Point A is at \((3, 7)\).
Reflection across the x-axis changes the sign of the y-coordinate.
The x-coordinate remains the same.
Point B has coordinates \((3, -7)\).

Choice A is incorrect because \((-3, 7)\) represents reflection across the y-axis, not the x-axis.

Question 17 - Correct Answer: A

The three given vertices are \((1, 1)\), \((1, 4)\), and \((4, 4)\).
From \((1, 1)\) to \((1, 4)\) is a vertical side of length 3.
From \((1, 4)\) to \((4, 4)\) is a horizontal side of length 3.
The square has side length 3.
The fourth vertex must be 3 units horizontally from \((1, 1)\) and 3 units vertically below \((4, 4)\).
The fourth vertex is \((4, 1)\).

Choice E is incorrect because \((3, 3)\) is the center of the square, not a vertex.

Question 18 - Correct Answer: B

Midpoint M is at \((2, 6)\) and endpoint P is at \((-1, 4)\).
Let endpoint Q be at \((x, y)\).
Using the midpoint formula: \(\frac{-1 + x}{2} = 2\) and \(\frac{4 + y}{2} = 6\).
From the first equation: \(-1 + x = 4\), so \(x = 5\).
From the second equation: \(4 + y = 12\), so \(y = 8\).
Endpoint Q is at \((5, 8)\).
Distance from P to Q: \(d = \sqrt{(5 - (-1))^2 + (8 - 4)^2}\).
\(d = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}\).
\(\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}\).

Choice A is incorrect because \(\sqrt{13}\) is the distance from M to either P or Q, which is half the distance from P to Q.

Question 19 - Correct Answer: A

A point in Quadrant II has negative x-coordinate and positive y-coordinate.
Reflection across the y-axis changes the sign of the x-coordinate.
The y-coordinate remains the same.
After reflection, the x-coordinate becomes positive and the y-coordinate remains positive.
The reflected point lies in Quadrant I.

Choice B is incorrect because reflecting across the y-axis moves the point to a different quadrant, not keeping it in Quadrant II.

Question 20 - Correct Answer: C

Points are A at \((1, 1)\), B at \((1, 5)\), and C at \((7, 5)\).
The right angle is at B.
Side AB is vertical from \((1, 1)\) to \((1, 5)\) with length \(5 - 1 = 4\).
Side BC is horizontal from \((1, 5)\) to \((7, 5)\) with length \(7 - 1 = 6\).
These are the two legs of the right triangle.
Area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\).
\(A = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12\).

Choice E is incorrect because 24 is the product of the two legs without dividing by 2.