Question

In triangle ABC, AB measures 25 cm and AC measures 35 cm.



The inequality
< s <
represents the possible third side length of the triangle, s, in centimeters.



The inequality
< p <
represents the possible values for the perimeter, p, of the triangle, in centimeters.

Answer 1

To find the possible lengths for the third side \( s \) of triangle ABC, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given:
- \( AB = 25 \, \text{cm} \)
- \( AC = 35 \, \text{cm} \)

Let \( s \) be the length of side \( BC \).

According to the triangle inequality:
1. \( AB + AC > BC \)
\[
25 + 35 > s \quad \Rightarrow \quad 60 > s \quad \Rightarrow \quad s < 60
\]
2. \( AB + BC > AC \)
\[
25 + s > 35 \quad \Rightarrow \quad s > 10
\]
3. \( AC + BC > AB \)
\[
35 + s > 25 \quad \Rightarrow \quad s > -10 \quad \text{(this is always true for positive } s)
\]

From the second inequality, the valid range for \( s \) becomes:
\[
10 < s < 60
\]

Next, we need to find the possible values for the perimeter \( p \) of triangle ABC:
\[
p = AB + AC + BC = 25 + 35 + s = 60 + s
\]

Now substituting the range for \( s \):
- When \( s = 10 \):
\[
p = 60 + 10 = 70
\]

- When \( s = 60 \):
\[
p = 60 + 60 = 120
\]

Thus, the possible range for the perimeter \( p \) is:
\[
70 < p < 120
\]

Summarizing:
- The inequality for the third side \( s \) is:
\[
10 < s < 60
\]

- The inequality for the perimeter \( p \) is:
\[
70 < p < 120
\]