The length of the side of a rhombus is 14 inches. The measure of each obtuse angle is 160 degrees. Find the length of the longer diagonal of the rhombus.
Since the rhombus has all sides of equal length, we know that each angle is the same. Since the obtuse angle is 160 degrees, the acute angle opposite of it is 180 - 160 = 20 degrees.
Because the opposite angles in a rhombus are equal, the other obtuse angle is also 160 degrees and the other acute angle is also 20 degrees.
To find the length of the longer diagonal, we can use the Law of Cosines. Let the length of the longer diagonal be d.
Using the Law of Cosines:
d^2 = 14^2 + 14^2 - 2(14)(14)cos(20)
d^2 = 196 + 196 - 392cos(20)
d^2 = 392 - 392cos(20)
Now, we need to solve for d:
d = √(392 - 392cos(20))
d = √(392 - 392(0.9397))
d = √(392 - 369.1144)
d = √22.8856
d ≈ 4.78
Therefore, the length of the longer diagonal of the rhombus is approximately 4.78 inches.