Two polygons are congruent and the perimeter of the first polygon is 38 cm. If the sides of the second polygon are consecutive integers (x, x+1, x+2, x+3, etc.), what value of x makes the polygons into congruent quadrilaterals? Use x as the smallest side.
Congruent figures have the same perimeters.
Quadrilaterals have four sides.
The perimeters of both would be 4x + 6.
Since 4x = 6 = 38, x = 8.
The side lengths are 8, 9, 10, and 11.
I don't see why the bring a second quadrilateral into the discusion.
To find the value of x that will make the two polygons congruent quadrilaterals, we need to set up an equation based on the given information.
Let's start by figuring out the perimeter of the second polygon. Since its sides are consecutive integers, we can add them up to find the sum.
The sum of consecutive integers can be found using the formula:
Sum = (n/2)(first term + last term)
In this case, the first term is x, and the last term will be (x + n - 1), where n is the total number of sides. Since we are dealing with a polygon, n will always be greater than or equal to 3.
So, the sum of the consecutive integers will be:
Sum = (n/2)(x + (x + n - 1))
The perimeter of the second polygon can then be expressed as:
Perimeter = Sum = (n/2)(2x + n - 1)
Since the two polygons are congruent, their perimeters must be equal. Therefore, we can set up the following equation:
38 = (n/2)(2x + n - 1)
Now, let's solve for x. First, we'll multiply both sides by 2 to eliminate the fraction:
76 = n(2x + n - 1)
Expanding the equation further, we get:
76 = 2nx + n^2 - n
Rearranging the terms:
n^2 + (2x - 1)n + 76 = 0
This is a quadratic equation in terms of n. We can now apply the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 2x - 1, and c = 76.
Substituting these values into the quadratic formula and simplifying, we get:
n = (1 - 2x ± √((2x - 1)^2 - 4*1*76)) / (2*1)
Simplifying further:
n = (1 - 2x ± √(4x^2 - 4x + 1 - 304)) / 2
n = (1 - 2x ± √(4x^2 - 4x - 303)) / 2
Since we're dealing with sides, n must be a positive integer. To satisfy this condition, the discriminant inside the square root (√(4x^2 - 4x - 303)) must be a perfect square.
To determine the smallest side x, we need to find out the value of x that satisfies this condition and makes the two polygons congruent quadrilaterals. We can do this by substituting integer values for x and checking if the discriminant is a perfect square.
Once we find a value of x that yields a perfect square discriminant, the two polygons will become congruent quadrilaterals.