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Pre-Algebra

The Coordinate Plane

Lesson

The coordinate plane is made of two number lines: the x-axis (horizontal) and the y-axis (vertical). They cross at the origin, where (0,0)(0, 0) sits.

A point is written as an ordered pair (x,y)(x, y), where xx tells you how far right (positive) or left (negative) and yy tells you how far up (positive) or down (negative).

The four quadrants

The plane is split into four regions called quadrants, numbered counter-clockwise starting from the upper right:

  • Quadrant I — upper right. x>0, y>0x > 0,\ y > 0
  • Quadrant II — upper left. x<0, y>0x < 0,\ y > 0
  • Quadrant III — lower left. x<0, y<0x < 0,\ y < 0
  • Quadrant IV — lower right. x>0, y<0x > 0,\ y < 0

Points on an axis (where x=0x = 0 or y=0y = 0) don’t belong to any quadrant.

Worked example 1

Which quadrant is (−3,5)(-3, 5) in?

x=−3x = -3 (negative — left of the y-axis) and y=5y = 5 (positive — above the x-axis). That puts it in Quadrant II.

Worked example 2

Find the distance between (−2,3)(-2, 3) and (4,3)(4, 3) (same y-coordinate — horizontal segment).

When the y-coordinates match, the distance is the absolute difference of the x-coordinates:

∣4−(−2)∣=∣6∣=6|4 - (-2)| = |6| = 6

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Quadrant of (4,5)?\text{Quadrant of } (4, 5)?

Problem 2

Quadrant of (−3,2)?\text{Quadrant of } (-3, 2)?

Problem 3

Quadrant of (−1,−8)?\text{Quadrant of } (-1, -8)?

Problem 4

Quadrant of (6,−2)?\text{Quadrant of } (6, -2)?

Practice

Standard problems matching the lesson.

Problem 5

Quadrant of (−7,3)?\text{Quadrant of } (-7, 3)?

Problem 6

Quadrant of (5,12)?\text{Quadrant of } (5, 12)?

Problem 7

Quadrant of (−4,−9)?\text{Quadrant of } (-4, -9)?

Problem 8

Quadrant of (8,−3)?\text{Quadrant of } (8, -3)?

Problem 9

x-coordinate of (5,−2)?\text{x-coordinate of } (5, -2)?

Problem 10

y-coordinate of (5,−2)?\text{y-coordinate of } (5, -2)?

Problem 11

Distance from (1,2) to (5,2)?\text{Distance from } (1, 2) \text{ to } (5, 2)?

Problem 12

Distance from (3,−1) to (3,4)?\text{Distance from } (3, -1) \text{ to } (3, 4)?

Problem 13

Distance from (−2,5) to (6,5)?\text{Distance from } (-2, 5) \text{ to } (6, 5)?

Problem 14

Distance from (4,−3) to (4,−8)?\text{Distance from } (4, -3) \text{ to } (4, -8)?

Problem 15

x-coordinate of the origin?\text{x-coordinate of the origin?}

Problem 16

y-coordinate of (−9,0)?\text{y-coordinate of } (-9, 0)?

Problem 17

Distance from (3,5) to (3,−2)?\text{Distance from } (3, 5) \text{ to } (3, -2)?

Problem 18

Quadrant of (−5,4)?\text{Quadrant of } (-5, 4)?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Quadrant of (−15,27)?\text{Quadrant of } (-15, 27)?

Problem 20

Distance from (−3,7) to (−3,−2)?\text{Distance from } (-3, 7) \text{ to } (-3, -2)?

Problem 21

Distance from (−5,4) to (8,4)?\text{Distance from } (-5, 4) \text{ to } (8, 4)?

Problem 22

Quadrant of a point with positive x and negative y?\text{Quadrant of a point with positive x and negative y?}

Problem 23

Midpoint x of (2,5) and (8,7)?\text{Midpoint x of } (2, 5) \text{ and } (8, 7)?

Problem 24

Midpoint y of (−1,4) and (3,10)?\text{Midpoint y of } (-1, 4) \text{ and } (3, 10)?

Problem 25

Total length of path (2,3)→(2,−4)→(−1,−4)\text{Total length of path } (2,3) \to (2,-4) \to (-1,-4)

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