the number of sides in 2 regular polygons is in the ratio 5:4 an the difference between their angles is 9^o. Find the number of sides in the polygon

Abcd

To find the number of sides in each regular polygon, let's set up an equation using the given information.

Let's assume the first regular polygon has x sides, where x is a multiple of 5, and the second regular polygon has y sides, where y is a multiple of 4.

Since the difference between their angles is 9 degrees, we can use the formula for the interior angle of a regular polygon:

Interior angle of a regular polygon = (180 * (n - 2)) / n

Let's set up the equation using the given information:

(180 * (x - 2)) / x - (180 * (y - 2)) / y = 9

Now, let's simplify and solve for x and y:

(180x - 360) / x - (180y - 360) / y = 9
[(180x - 360)y - (180y - 360)x] / xy = 9
(180xy - 360y - 180xy + 360x) / xy = 9
(360x - 360y) / xy = 9
360x - 360y = 9xy
360(x - y) = 9xy
40(x - y) = xy

Since x and y are both multiples of 4 and 5 respectively, we can express them as:

x = 4a
y = 5b

Substituting these values in:
40(4a - 5b) = (4a)(5b)
160a - 200b = 20ab
8a - 10b = ab

Since a and b are both integers, the left-hand side (8a - 10b) must be divisible by ab. The only way this is possible is if (8a - 10b) is divisible by both a and b separately.

To find the possible values for a and b, we can use trial and error. Let's start with a = 10 and b = 8:

8 * 10 - 10 * 8 = ab
80 - 80 = 10 * 8
0 = 80 (No solution, let's try other values)

Let's try a = 5 and b = 4:

8 * 5 - 10 * 4 = ab
40 - 40 = 5 * 4
0 = 20 (No solution, let's try other values)

Let's try a = 4 and b = 5:

8 * 4 - 10 * 5 = ab
32 - 50 = 4 * 5
-18 = 20 (No solution, let's try other values)

Let's try a = 8 and b = 10:

8 * 8 - 10 * 10= ab
64 - 100 = 8 * 10
-36 = 80 (No solution, let's try other values)

From our trial and error, we can see that there's no solution for a and b such that (8a - 10b) is divisible by ab.

Therefore, there is no possible combination of regular polygons with their number of sides in a ratio of 5:4 and a difference between their angles is 9 degrees.

The internal angle each corner of a regular polygon is 180 - 360/n degrees.

Let N be the number of sides of polygon with fewer sides. The other polygon then has 5N/4 sides.

180 - 360/N - [180 - 360*0.8/N] = -9
-360/N + 288/N = -9
72/N = 9
N = 8
The other polygon has 10 sides.
The internal angles are 135 and 144 degrees.