IMPORTANT RULE: If two sides of a triangle have lengths A and B, then . . .
the DIFFERENCE between A and B < length of the third side < the SUM of A and B

So, for ∆ABC, we know the lengths of two sides, but we don't know the length of side BC
Applying the above rule, we see that: 9 - 5 < length of side BC < 9 + 5
Simplify, to get: 4 < length of side BC < 14

So, for example, it COULD be the case that BC = 10, in which case, we have this:

When we apply the above rule to ∆BCD, we see that: 10 - 10 < x < 10 + 10
In other words, 0 < x < 20

This means x COULD equal 3, and x COULD equal 15
So, i and ii are both possible.

What about iii?

Let's see what happens when we try to MAXIMIZE the value of x.

To do so, we'll first maximize the value of side BC.
Since we already know that 4 < length of side BC < 14
So, the biggest possible value of BC is just a little bit less than 14.

Let's see what happens when side BC has length 13.999999 (which is a bit less than 14)

When we apply the above rule to ∆BCD, we see that: 13.999999 - 10 < x < 13.999999 + 10
In other words, 3.999999 < x < 23.999999

Since side BC must be less than 14, we can see that x must be less than 24
In other words, x cannot equal 24
So, statement iii is not possible

Answer: B

Cheers,
Brent

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