Final answer:
Without additional information regarding the specific attributes of triangle XYZ, particularly whether it is right-angled and the position of WZ, we cannot accurately calculate the length of XW.
Step-by-step explanation:
To find the length of XW to the nearest integer, we can use the properties of right-angled triangles and trigonometry. In triangle XYZ, the measures of angles X and Y are given as 40° and 45°, respectively. Since the sum of angles in a triangle is 180°, we can find angle Z by subtracting the sum of angles X and Y from 180°, which gives us 180° - 40° - 45° = 95° for angle Z.
However, this information alone does not help us find XW directly because it is not given that triangle XYZ is right-angled or that line WZ is the height from angle X to the hypotenuse YZ in triangle XYZ. Since additional information is required such as whether triangle XYZ is right-angled or not or the relationship to angle WZ, we cannot determine the length of XW.
If, for example, it were stated that angle Y was a right angle, we could have applied trigonometry using the sine of angle X to find the length of XW. In this case, we would use the formula:
XW = XY * sin(X), which would turn to XW = 12 * sin(40°). After calculating this we would get the length of XW in decimal form and we could round it to the nearest integer.
Due to the lack of specific details in the question, we are unable to provide an accurate answer. Thus, the question is missing some crucial information needed to solve for the length of XW.