jmc

Question

In the triangle shown, for A to be the largest angle of the triangle, it must be that m<x<n. What is the least possible value of n-m, expressed as a common fraction? FIGURE

Accepted Answer

The sides of the triangle must satisfy the triangle inequality, so

A B + A C > B C

,

A B + B C > A C

, and

A C + B C > A B

. Substituting the side lengths, these inequalities turn into

(x + 4) + (3 x) (x + 4) + (x + 9) (3 x) + (x + 9) > x + 9, > 3 x, > x + 4,

which give us

x > 5/3

,

x < 13

, and

x > - 5/3

, respectively.

However, we also want

∠ A

to be the largest angle, which means that

B C > A B

and

B C > A C

. These inequalities turn into

x + 9 > x + 4

(which is always satisfied), and

x + 9 > 3 x

, which gives us

x < 9/2

.

Hence,

x

must satisfy

x > 5/3

,

x < 13

,

x > - 5/3

, and

x < 9/2

, which means

\frac{5}{3} < x < \frac{9}{2} .

The answer is

9/2 - 5/3 = \frac{17}{6}

.

(Also, note that every value of

x

in this interval makes all the side lengths positive.)