Final answer:
Using the Triangle Inequality Theorem, the lengths that cannot be the third side of a triangle with sides of 22 inches and 10 inches are 10 inches, 22 inches, and 37 inches.
Step-by-step explanation:
To determine which lengths cannot be the length of the third side of a triangle when the other two sides are 22 inches and 10 inches, 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. To apply this to the given problem, we'll test each potential third side length.
- For side A (10 inches): 22 + 10 > 10 and 22 + 10 > 22, but 10 + 10 is not greater than 22, so A is not possible.
- For side B (32 inches): 22 + 10 > 32, so B is possible.
- For side C (15 inches): 22 + 10 > 15 and 22 + 15 > 10, and 10 + 15 > 22, so C is possible.
- For side D (22 inches): 22 + 10 is not greater than 22, so D is not possible.
- For side E (12 inches): 22 + 10 > 12 and 22 + 12 > 10, and 10 + 12 > 22, so E is possible.
- For side F (20 inches): 22 + 10 > 20 and 22 + 20 > 10, and 10 + 20 > 22, so F is possible.
- For side G (37 inches): 22 + 10 is not greater than 37, so G is not possible.
- For side H (12.2 inches): 22 + 10 > 12.2 and 22 + 12.2 > 10, and 10 + 12.2 > 22, so H is possible.
Therefore, the lengths that cannot be on the third side of the triangle are 10 inches (A), 22 inches (D), and 37 inches (G).