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Answer:

The length of the median (AB) ̅ of trapezoid FLOP be m.

Given, m = 2x + 4, where x is a number.

Also, (FL) ̅ is four more than a number, so let (FL) ̅ = n + 4, where n is the same number.

And, (PO) ̅ is twice a number and ten, so let (PO) ̅ = 2x + 10.

Since (AB) ̅ is the median, it divides the trapezoid into two equal areas. Let the height of the trapezoid be h.

Then, the length of the parallel sides of the trapezoid can be found using the Pythagorean theorem:

(FP) ̅^2 = (FL) ̅^2 - h^2

and

(OP) ̅^2 = (PO) ̅^2 - h^2

Since (FP) ̅ = (OP) ̅ = m/2, we can substitute these values in the above equations:

(m/2)^2 = (n + 4)^2 - h^2

and

(m/2)^2 = (2x + 10)^2 - h^2

Simplifying these equations, we get:

m^2/4 = n^2 + 8n + 16 - h^2

and

m^2/4 = 4x^2 + 40x + 100 - h^2

Equating the two equations, we get:

n^2 + 8n + 16 - h^2 = 4x^2 + 40x + 100 - h^2

Simplifying further, we get:

n^2 + 8n + 16 = 4x^2 + 40x + 100

Rearranging terms, we get:

4x^2 + 40x - n^2 - 8n - 84 = 0

Now we can solve for x using the quadratic formula:

x = (-40 ± sqrt(40^2 - 4(4)(-n^2 - 8n - 84))) / (2(4))

Simplifying this equation, we get:

x = (-10 ± sqrt(n^2 + 8n + 31)) / 2

Finally, we can substitute the value of x in the equation for m:

m = 2x + 4

to get the length of the median (AB) ̅ in terms of n:

m = -5 ± sqrt(n^2 + 8n + 31) + 4

m = -1 ± sqrt(n^2 + 8n + 31)

Therefore, the mathematical equations are:

m = 2x + 4

(FL) ̅ = n + 4

(PO) ̅ = 2x + 10

n^2 + 8n + 16 = 4x^2 + 40x + 100

x = (-10 ± sqrt(n^2 + 8n + 31)) / 2

m = -1 ± sqrt(n^2 + 8n + 31)

...(Answer By: John Isiah A. Dangase)