Two polygons have te number of their sides in the ratio 1:3.The sums of their interior angles are in the ratio of 4;1. Find the number of sides of each polygon.
Hm... quite a complicate question. A less complicate way to do this is trial and error. You can make a table with "n" sides and the sum of the angles in degree for each polygon (ie. triangle (3 sides) = 180, 4 sides = 360, 5 sides = 540... and so on). And base on the values, you can find the matching pairs that fit the ratio criteria. But I guess that's what your teacher intend you to do. So let's try to figure this out mathematically:
You can come up with a formula to calculate the sum of the angles from the small table you created above:
Angle = 180n - 360.
Now you know that the ratio of the angles is 4:1 and the sides is 3:1, and so you have another formula for the other polygon:
4(Angle) = (3n)180 - 360.
Angle = 135n - 90
Now using substituion and plug it back into the first equation, you get:
135n - 90 = 180n - 360
270 = 45n
n = 6
Now that you know that one polygon has 6 sides, the other one must have three time as many, thus 18 sides. Check the sum of the angles for each, and you should come up with a ratio of 1:4 on the angles for both polygons. Problem solved!
I meant: NOT what your teacher intend you to do. Sorry for mistype. :p
You surely meant that the sum of their angle sum is 1:4 , not 4:1
let the number of sides of the first polygon be n
then the numer of sides of the second is 3n
(notice n : 3n = 1 : 3 )
angle sum of first = 180(n-2)
angle sum of second = 180(3n-2)
so,
180(n-2)/(180(3n-2)) = 1/4
(n-2)/(3n-2) = 1/4
4n-8 = 3n-2
n = 6
first polygon has 6 sides,
the second has 18 sides
To solve this problem, let's assume that the first polygon has x sides, and the second polygon has y sides.
According to the given information, the ratio of the number of sides of the first polygon to the number of sides of the second polygon is 1:3. This can be written as:
x:y = 1:3 (Equation 1)
Now, let's analyze the second piece of information. The sum of the interior angles of a polygon is given by the formula: (n-2) * 180, where n is the number of sides of the polygon.
So, the sum of the interior angles of the first polygon would be (x-2) * 180, and the sum of the interior angles of the second polygon would be (y-2) * 180.
According to the given information, the ratio of the sums of the interior angles of the two polygons is 4:1. This can be written as:
(x-2) * 180 : (y-2) * 180 = 4:1 (Equation 2)
Now, let's simplify Equation 2 by dividing both sides of the equation by 180:
(x-2) : (y-2) = 4:1 (Equation 3)
Next, let's cross-multiply Equation 3 to eliminate the fractions:
1 * (x-2) = 4 * (y-2)
x-2 = 4y-8
x = 4y-6 (Equation 4)
Now, we can substitute Equation 4 into Equation 1:
4y-6:y = 1:3
3(4y-6) = y
12y - 18 = y
11y = 18
y = 18/11
Since the number of sides of a polygon must be a whole number, y cannot be a fraction. Therefore, there is no solution to this problem.
In conclusion, there are no polygons that satisfy the given conditions.