Question

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

Answer 1

Answer:

The solution for s is the interval--------> $(2,18)$

All real numbers greater than $2$ and less than $18$

Step-by-step explanation:

we know that

The Triangle Inequality Theorem, states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side

so

Let

s-----> the third length side of the given triangle

Applying the triangle inequality theorem

$8+10> s$

$18> s$ -------> $s$

$8+s> 10$

$s> 10-8$

$s> 2\ cm$

The solution for s is the interval--------> $(2,18)$

All real numbers greater than $2$ and less than $18$

Answer 2

The third side of any triangle must be greater than the difference of the given two sides and must be less than the sum of the given two sides.

Given 8cm and 10cm.

The third side s,

s > (10 - 8)                                 s < (10 + 8)
s > 2                                            s < 18

s > 2   and s < 18 .         

Combining the two inequalities:

2 <  s < 18                    Range is between 2 and 18.

So the range for third side is a number between 2 and 18  cm.      Note that 2 and 18 are not included.

s could be any of 3, 4, 5, 7, 9, 13,..17.  I hope this helps.