Practice Coordinate Representation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Describing geometric objects precisely using ordered pairs (x,y)(x, y) or triples (x,y,z)(x, y, z) in a coordinate system.

Every point has a unique numerical 'address' like (3,4)(3, 4) that locates it exactly on the plane.

Showing a random 20 of 50 problems.

Example 1

challenge
Why does adding a zz-coordinate let you represent objects that 2D coordinates cannot?

Example 2

challenge
Prove that the triangle with vertices (0,0)(0,0), (6,0)(6,0), (3,33)(3, 3\sqrt{3}) is equilateral using coordinates.

Example 3

medium
Why does coordinate geometry let you prove geometric facts with algebra?

Example 4

challenge
Find the shortest distance from the point (4,5)(4, 5) to the line 3x+4yโˆ’12=03x + 4y - 12 = 0.

Example 5

easy
Which point is in Quadrant IV: (โˆ’1,โˆ’1)(-1, -1), (3,โˆ’5)(3, -5), or (0,4)(0, 4)?

Example 6

hard
Find the distance from (1,2,3)(1, 2, 3) to the origin in 3D.

Example 7

medium
Does (2,โˆ’3)(2, -3) lie on the line y=โˆ’2x+1y = -2x + 1?

Example 8

challenge
Convert the Cartesian point (โˆ’2,2)(-2, 2) to polar coordinates with r>0r > 0 and 0โ‰คฮธ<2ฯ€0 \le \theta < 2\pi.

Example 9

hard
Show that the points (0,0),(4,0),(4,3),(0,3)(0, 0), (4, 0), (4, 3), (0, 3) form a rectangle (not just a parallelogram).

Example 10

easy
Which point is in the second quadrant: (2,3)(2, 3), (โˆ’2,3)(-2, 3), or (โˆ’2,โˆ’3)(-2, -3)?

Example 11

easy
Does the point (4,0)(4, 0) lie on the xx-axis or the yy-axis?

Example 12

medium
Write the equation of the vertical line through (7,2)(7, 2).

Example 13

easy
What is the equation of the horizontal line through (3,โˆ’4)(3, -4)?

Example 14

easy
How many coordinates locate a point in 2D space?

Example 15

challenge
Show that the points (0,0)(0,0), (4,0)(4,0), (4,3)(4,3), (0,3)(0,3) form a rectangle, and find its area, using coordinates.

Example 16

easy
Does the point (โˆ’3,4)(-3, 4) lie on the circle x2+y2=25x^2 + y^2 = 25?

Example 17

medium
A point in 3D is (2,5,9)(2, 5, 9). What is its projection onto the xyxy-plane, in coordinates?

Example 18

easy
In the point (3,5)(3, 5), which number is the yy-coordinate?

Example 19

medium
Find the yy-intercept of the line 3x+2y=123x + 2y = 12.

Example 20

medium
Write the equation of the circle with center (3,โˆ’4)(3, -4) and radius 55.
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