Final answer:
The range for the third side, x, of an acute triangle with sides 4 cm and 8 cm is more than 4 cm but less than 8 cm (4 < x < 8 cm) to satisfy the triangle inequality theorem and maintain the triangle as acute.
Step-by-step explanation:
The question is asking us to find the range of possible values for the third side, x, of an acute triangle when the other two sides measure 4 cm and 8 cm. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we can set up two inequalities: x + 4 > 8 (which simplifies to x > 4) and x + 8 > 4 (which simplifies to x > -4, but this is not useful since sides of a triangle cannot be negative). Similarly, the difference between the lengths of any two sides must be less than the length of the third side, so we can set up another inequality: 8 - 4 < x (which simplifies to x > 4). Combining these, we find that the length of x must be greater than 4 cm but less than 12 cm (as 4 + 8), meaning the best choice is 4 < x < 12 cm, which is not one of the options provided. However, since we are discussing an acute triangle, x must also be less than 8 cm to prevent the triangle from becoming obtuse.
Thus, the correct option that best represents the range of values for the third side, x, of an acute triangle would be option d) 4≤x≤8 cm.