Let ABCDEF be an equiangular hexagon with perimeter 98 such that EF=2CD=4AB and BC=2FA. What is the length of EF?
To solve this problem, we need to rely on the properties of an equiangular hexagon. An equiangular hexagon is a hexagon with all interior angles equal to 120 degrees.
Let's start by considering the given conditions:
1. EF = 2CD
2. EF = 4AB
3. BC = 2FA
4. Perimeter = 98
Since we know the hexagon is equiangular, we can determine the measures of each angle by dividing the sum of the angles of a hexagon (720 degrees) by 6. Each angle in this equiangular hexagon will measure 120 degrees.
Now, let's set up an equation to find the length of each side using the given conditions:
Perimeter of the hexagon = AB + BC + CD + DE + EF + FA
Since all the sides are equal, we can substitute:
AB + BC + CD + DE + EF + FA = 6AB
AB + AB + AB + AB + 4AB + AB = 6AB
9AB = 98 (using the given perimeter of 98)
Dividing both sides by 9:
AB = 98 / 9
Now, we can use the relationship EF = 4AB to find the length of EF:
EF = 4 * AB
EF = 4 * (98 / 9)
Simplifying:
EF = (4 * 98) / 9
EF = 392 / 9
Therefore, the length of EF is approximately 43.556.
opposite sides must be equal, so if AB=x, we have
2(x)+2(2x)+2(4x) = 98
7x = 49
x = 7
EF = 4*7 = 28