# SAT Math Multiple Choice Practice Question 494: Answer and Explanation

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**Question: 494**

**14.** Triangle *ABC* has a perimeter of 10, and the lengths of its sides are all integers. If *a* is the length of side , what is the difference between the largest and smallest possible values of *a* ?

A. 1

B. 2

C. 3

D. 4

E. 7

**Correct Answer:** B

**Explanation:**

**B** The "Third Side Rule" states that any side must be less than the sum of the other two sides, and greater than the difference between the other two sides. In other words, if the sides are *a*, *b*, and *c*, |*b* – *c*| < *a* < *b* + *c*. Find the smallest possible value of *a*. If *a* = 1, then *b* + *c* = 9. For example, *b* = 4 and *c* = 5. This is illegal, as |*b* – *c*| = *a*. If *a* = 2, then *b* = 4 and *c* = 4, so 2 is okay. The largest *a* can be is 4, because if *a* = 5, then *a* = *b* + *c* and you have violated the Third Side Rule. Thus, the difference between the largest possible value of *a* and the smallest possible value of *a* is 4 – 2 = 2.