Answer:
0.5 < x < 16.5
Step-by-step explanation:
For a triangle (by the triangular inequality theorem), we know that the sum of any two sides must be larger than the other side.
When we have a triangle and we know two sides, let's define the measures of these sides as S1 and S2. (such that S2 > S1)
By the first property, we will have that S1 + S2 > S3
Where S3 is the third side, the one that we do not know.
But there is also a lower restriction, given by:
S2 - S1 < S3 < S1 + S2
This is because we also must have:
S1 + S3 > S2
and
S2 + S3 > S1
We can rewrite both of these to get:
S2 - S1 < S3
S1 - S2 < S3
Because S2 > S1, the first inequality is more restrictive, so we need to use that one.
Then we get the inequality:
S2 - S1 < S3 < S1 + S2
Ok, in this case, the shorter side is 8.0 then:
S1 = 8.0
And the longer side is 8.5, then:
S2 = 8.5
And the third side is x, S3 = x
Replacing those in our inequality, we get:
8.5 - 8.0 < x < 8.5 + 8.0
0.5 < x < 16.5
This is the range of possible sizes for side x.
where r represents the radius.