Practice Distance in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The length of the shortest path between two points, always a non-negative real number.

'As the crow flies'—the straight-line separation between two locations.

Showing a random 20 of 50 problems.

Example 1

medium
Extend the distance formula to 3D: find the distance between (1,2,3)(1, 2, 3) and (4,6,15)(4, 6, 15).

Example 2

hard
Find the distance from the origin to the midpoint of the segment joining (2,6)(2, 6) and (8,2)(8, 2).

Example 3

hard
A circle passes through (0,0),(6,0),(0,8)(0, 0), (6, 0), (0, 8). Find the center and radius using equidistance.

Example 4

medium
Show that the points (0,0),(5,0),(5,5)(0,0), (5,0), (5, 5) form a right isosceles triangle. Compute all side lengths.

Example 5

hard
Find the distance between the parallel lines y=2x+1y = 2x + 1 and y=2x−3y = 2x - 3.

Example 6

challenge
Three towns sit at (0,0)(0,0), (8,0)(8,0), and (4,3)(4, 3). Is the third town closer to the first or the second?

Example 7

medium
Is the triangle with vertices (0,0),(5,0),(0,12)(0,0), (5,0), (0, 12) a right triangle? Verify with side lengths.

Example 8

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

Example 9

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

Example 10

medium
Find the distance between (1,1)(1, 1) and (5,4)(5, 4).

Example 11

challenge
Find the point on the xx-axis equidistant from A(1,2)A(1, 2) and B(5,4)B(5, 4).

Example 12

medium
A circle has center (2,1)(2, 1) and passes through (5,5)(5, 5). Find its radius.

Example 13

challenge
Prove that the distance function d(A,B)d(A, B) on the plane satisfies d(A,B)=0⇔A=Bd(A, B) = 0 \Leftrightarrow A = B, symmetry, and the triangle inequality.

Example 14

hard
Let A=(0,0)A = (0,0) and B=(10,0)B = (10, 0). Find the locus of points PP such that PA=2⋅PBPA = 2 \cdot PB.

Example 15

medium
Find the distance between (−3,−1)(-3, -1) and (1,2)(1, 2).

Example 16

challenge
Find the minimum possible distance from a point on the line x+y=10x + y = 10 to the origin.

Example 17

medium
Extend the distance formula to 3D: find the distance between (0,0,0)(0,0,0) and (2,3,6)(2,3,6).

Example 18

medium
Is triangle with vertices (0,0)(0,0), (4,0)(4,0), (0,3)(0,3) a right triangle? Use distances.

Example 19

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

Example 20

hard
Three points A(0,0),B(4,0),C(2,h)A(0,0), B(4,0), C(2, h) form an equilateral triangle. Find hh.
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