Final answer:
To calculate the measure of the internal angle of the largest polygon, we need to determine the number of sides in the largest polygon. Given the relation between the total diagonals of the two polygons, we can find the number of sides using a formula. By setting up an equation based on the given information, we can solve for the measure of the internal angle of the largest polygon as 90 degrees.
Step-by-step explanation:
To calculate the measure of the internal angle of the largest polygon, we need to determine the number of sides in the largest polygon. Since the relation between the total diagonals of the two regular polygons is infinite, we can determine the number of sides in the polygons using the formula n(n-3)/2, where n is the number of sides. Let's assume the smaller polygon has 'a' sides and the larger polygon has 'b' sides.
Given that the central angles are in the relation 1/4, we know that the smaller polygon has an internal angle of 360/a degrees and the larger polygon has an internal angle of 360/b degrees.
Now, we can set up the following equation based on the information given:
360/a = (1/4) * 360/b
Using cross-multiplication, we can simplify the equation to:
4ab = b
Solving for 'b', we get:
b = 4
Therefore, the measure of the internal angle of the largest polygon is 360/4 = 90 degrees.