Practice Distance Formula in Math

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

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Quick Recap

A formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

Imagine two points on a grid. Draw a horizontal line from one and a vertical line from the other to form a right triangle. The horizontal leg is the difference in xx-coordinates, the vertical leg is the difference in yy-coordinates, and the hypotenuse—the direct distance—comes from the Pythagorean theorem. The distance formula is just a2+b2=c2a^2 + b^2 = c^2 in coordinate clothing.

Showing a random 20 of 50 problems.

Example 1

medium
The equation (x−3)2+(y−2)2=5\sqrt{(x-3)^2 + (y-2)^2} = 5 describes what?

Example 2

easy
Find the distance between (2,3)(2,3) and (2,9)(2,9).

Example 3

medium
Show that the triangle with vertices A(0,0)A(0, 0), B(3,4)B(3, 4), and C(6,0)C(6, 0) is isosceles.

Example 4

medium
Find the points on the xx-axis that are 55 units away from the point (2,4)(2, 4).

Example 5

easy
Find the distance between (0,0)(0,0) and (3,4)(3,4).

Example 6

hard
If (x,y)(x, y) is equidistant from (0,0)(0, 0) and (6,8)(6, 8), find the equation relating xx and yy.

Example 7

easy
Find the distance between (0,0)(0, 0) and (9,12)(9, 12).

Example 8

easy
What is the distance formula between points (x1,y1)(x_1,y_1) and (x2,y2)(x_2,y_2)?

Example 9

hard
The midpoint of segment ABAB is (3,5)(3, 5) and A=(−1,2)A = (-1, 2). Find BB and the length of ABAB.

Example 10

challenge
Explain why the distance formula gives the same result as the Pythagorean theorem, and what 'distance' means when the two points coincide.

Example 11

medium
A square has vertices (0,0),(a,0),(a,a),(0,a)(0,0), (a, 0), (a, a), (0, a). Find the length of its diagonal in terms of aa.

Example 12

medium
Find the distance in 3D between (1,2,2)(1,2,2) and (4,6,14)(4,6,14).

Example 13

medium
Find the distance between (1,−2)(1, -2) and (4,2)(4, 2).

Example 14

medium
Find the distance between (1,2)(1,2) and (4,8)(4,8), leaving the answer in simplest radical form.

Example 15

medium
Find the distance between (−5,2)(-5, 2) and (3,−4)(3, -4), in simplest radical form.

Example 16

medium
A point (x,0)(x,0) on the x-axis is equidistant from (0,0)(0,0)... actually, find x so (x,0)(x,0) is distance 10 from (6,8)(6,8).

Example 17

medium
Show that the points A(1,1),B(4,5),C(8,2)A(1, 1), B(4, 5), C(8, 2) form a right triangle.

Example 18

easy
Why does the order of subtraction not matter in the distance formula?

Example 19

easy
Find the distance between (0,0)(0,0) and (5,12)(5,12).

Example 20

medium
Find the distance between (2,−3)(2,-3) and (7,9)(7,9).
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