Final answer:
The third side of the triangle must be greater than 8 cm and less than 26 cm. The possible lengths for the third side from the given options are 9 cm, 15 cm, and 21 cm.
Step-by-step explanation:
To determine all possible values for the third side of a triangle with two sides measuring 17 cm and 9 cm, we use the triangle inequality theorem. This theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Similarly, the difference of the lengths of any two sides must be less than the length of the third side.
Let's denote the length of the third side as x. To find the possible values of x, we set up the following inequalities:
- 17 + 9 > x => 26 > x
- 17 + x > 9 => x > -8 (which is always true since a side length cannot be negative)
- 9 + x > 17 => x > 8
Hence, the third side x must be greater than 8 cm and less than 26 cm. We can now look at the given options:
- (A) 9 cm - This is possible since 9 > 8 and 9 < 26.
- (B) 15 cm - This is possible since 15 > 8 and 15 < 26.
- (C) 21 cm - This is possible since 21 > 8 and 21 < 26.
- (D) 26 cm - This is not possible since it is not less than 26.
- (E) 27 cm - This is not possible since it is greater than 26.
The correct answers are options (A), (B), and (C).