What is the area if a regular hexagon if the distance from a midpoint of a side to the midpoint of the opposite side is 10

50√3 or  86.6

Step-by-step explanation:

Divide the hexagon into six equilateral triangles as in Figure 1 below.

Then draw a line from the midpoint of one side to the midpoint of the opposite side.

The height (h) of each triangle will be 5.

To find the area of the whole triangle, we first need to find the length of the base (see Figure 2).

If the base length (b) is 2s, we have the relation

5/s = tan60°

5/s = √3      Multiply each side by s

5 = s√3        Divide each side by √3

s = 5/√3       Rationalize

s = (5√3)/3

b = 2s

b = (10√3)/3

The area (A) of the triangle is

A = ½bh

A = ½ × (10√3)/3 × 5

A = (25√3)/3

There are six equilateral triangles in the hexagon, so

Total area = 6A

Total area = 6 × (25√3)/3

Total area = 50√3

Total area ≈ 86.6  

step-by-step explanation:

this is not possible to solve with the info given.   you need at least an one angle somewhere.