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 $|x - y|?$

$2$

$4$

$6$

$8$

$10$

Answer: D

Concepts:triangle inequality bounding to limit cases

Difficulty rating: 1370

Solution:

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.

2000 AMC 10 Problem 10

Answer: D

Difficulty rating: 1370

Solution:

Problem 10 in Other Years