A triangle has two sides with lengths 7 and 12 which of the folowing lengths could represent the third side

If s₃ is the length of the 3rd side, the possible values are in the interval.

5 < s₃ < 19

Step-by-step explanation:

For a triangle, we know that the sum of any two sides is always larger than the other side.

So if a triangle has 3 sides:

s₁, s₂, and s₃, we have:

s₁ + s₂ > s₃

s₁ + s₃ > s₂

s₂ + s₃ > s₁

In this case, we know that two sides are:

s₁ = 7

s₂ = 12

And we want to find the possible values of s₃.

Then if we use the above inequalities, we get:

7 + 12 > s₃

7 + s₃ > 12

12 + s₃ > 7

With the first one we get:

19 > s₃

Now we can rewrite the other two as:

s₃ > 12 - 7 = 5

s₃ > 7 - 12 = -5

The first one is more restrictive than the second:

s₃ > 5 > -5

So we can use only the first one.

Then the two inequalities are

s₃ > 5

s₃ < 19

Then the range is:

5 < s₃ < 19

This means that any value between 5 and 19 can be a possible length of the third side (5 and 19 are not possible lengths)

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