2000 AMC 10 Problem 10

Question

The sides of a triangle with positive area have lengths

4,

6,

and

x.

The sides of a second triangle with positive area have lengths

4,

6,

and

y.

What is the smallest positive number that is not a possible value of

|x - y|?

2

4

6

8

10

The triangle inequality forces each of

x

and

y

to lie strictly between

2

and

10

Find the range of

|x - y|

when

x

and

y

each range over that open interval

By the triangle inequality, each of

x

and

y

can be any number strictly between

6 - 4 = 2

and

6 + 4 = 10.

Then

|x - y|

can take any value with

0 \le |x - y| \lt 8.

The smallest positive number not attainable is

10 - 2 = 8.

Thus, the correct answer is D.

Problem 10 in other years