An acute triangle has two sides measuring 8 cm and 10 cm . what is the best representation of the possible range of the values for the third side s?

the best interval that satisfies all the conditions is ]10,12.8[

Explanation:
Assume that the sides of the triangle are:
a = 8
b = 10
c = s
We are given that s is the longest side

For the triangle to be acute:
c^2 < a^2 + b^2

Substitute with the values of a, b and s in the above inequality and solve for s as follows:
c^2 < a^2 + b^2
s^2 < (8)^2 + (10)^2
s^2 < 164

We will solve for the zero condition. This means that we will solve for s^2=164 as follows:
s^2=164
s = + or - √164
either s = 12.8
or s = -12.8

Since the condition is that s < √164, this means that s < 12.8
So, initially the range is ]-∞, 12.8[

We are also given that s is the longest side, this means that s should be greater than 10.
The range now becomes:
]10,12.8[

One final condition to check. In any triangle, the sum of any two sides should be greater than the third.
Therefore:
8+10 > s
18 > s > satisfied in our range

Based on the above, the best interval that satisfies all the conditions is ]10,12.8[

Hope this helps :)

P(chicago, illinois)