Final answer:
Applying the Pythagorean theorem, the triangle with sides of lengths 4 cm, 9 cm, and 10 cm is not a right triangle because 4² + 9² (which equals 97) does not equal 10² (which is 100).
Step-by-step explanation:
To determine if a triangle with sides of lengths 4 cm, 9 cm, and 10 cm is a right triangle, we use the Pythagorean theorem. According to this theorem, the square of the hypotenuse's length—the side that faces the right angle—in a right triangle equals the sum of the squares of the lengths of the other two sides. The formula is represented as a² + b² = c², where c is the hypotenuse and a and b are the other two sides of the triangle.
Let's apply this formula to our triangle with sides 4 cm, 9 cm, and 10 cm:
- For sides 4 cm (a) and 9 cm (b), the sum of their squares is 4² + 9² = 16 + 81 = 97.
- The square of the longest side, 10 cm (c), is 10² = 100.
Since 97 does not equal 100, the triangle with sides of lengths 4 cm, 9 cm, and 10 cm is not a right triangle.