Question

applying the triangle inequality theorem
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.

Explanation:

Step1: Apply Triangle Inequality for Side \( s \)

The Triangle Inequality Theorem states that the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. For sides \( AB = 25 \) cm, \( AC = 35 \) cm, and third side \( s \):
\( |35 - 25| < s < 35 + 25 \)
Simplify: \( 10 < s < 60 \).

Step2: Find Perimeter Range

The perimeter \( p = AB + AC + s = 25 + 35 + s = 60 + s \).
Substitute \( s \) from Step 1:

• Minimum \( p \): \( 60 + 10 = 70 \)
• Maximum \( p \): \( 60 + 60 = 120 \)

Thus, \( 70 < p < 120 \).

Answer:

For the third side \( s \): \( 10 < s < 60 \)
For the perimeter \( p \): \( 70 < p < 120 \)

(So the first inequality blanks are \( 10 \) and \( 60 \); the second inequality blanks are \( 70 \) and \( 120 \).)