A rhombus has base angles of 60 and 120. The length of each side is 16. What is the area of the rhombus?
To find the area of a rhombus, you can use the formula:
Area = (diagonal1 * diagonal2) / 2
However, since the length of the diagonals is not given, we need to calculate them using the given information.
In a rhombus, the diagonals are perpendicular bisectors of each other, and they form four congruent right-angled triangles.
Given that the base angles of the rhombus are 60 and 120 degrees, we can determine the lengths of the diagonals.
Let's label the diagonals as d1 and d2.
In a rhombus, each interior angle is supplementary to its adjacent angle, so:
Angles 1 and 2 = 180 - 60 = 120 degrees
Now, we can use the law of cosines to find the lengths of the diagonals. The law of cosines states that, for any triangle with sides a, b, and c, and opposite angle C:
c^2 = a^2 + b^2 - 2*a*b*cos(C)
In the given triangle, we know the two sides (16) and the angle opposite to the side we want to find (120 degrees):
d1^2 = 16^2 + 16^2 - 2*16*16*cos(120)
Calculating the above equation, we get:
d1^2 = 512
Taking the square root of both sides, we find:
d1 = √512 = 16√2
Similarly, for the other diagonal, we can use the law of cosines with angles 1 and 2 (120 degrees):
d2^2 = 16^2 + 16^2 - 2*16*16*cos(120)
Again, calculating this equation, we have:
d2^2 = 512
Taking the square root of both sides, we get:
d2 = √512 = 16√2
Now that we know the lengths of both diagonals, we can calculate the area using the formula mentioned earlier:
Area = (d1 * d2) / 2
Area = (16√2 * 16√2) / 2
Area = (16 * 16 * 2) / 2
Area = 256 square units
Therefore, the area of the rhombus is 256 square units.
To find the area of a rhombus, you can use the formula: Area = (diagonal1 * diagonal2) / 2.
In this case, we don't know the length of the diagonals, but we have some information that can help us find them. Since the base angles of the rhombus are 60 and 120 degrees, we can split the rhombus into two congruent triangles.
Since the two triangle's angles are 90 degrees, 60 degrees, and 30 degrees, we have a 30-60-90 triangle. In a 30-60-90 triangle, the ratio of the side lengths is 1:√3:2.
Given that the length of each side of the rhombus is 16, we can calculate the length of each diagonal by using the side length and the ratio of the side lengths in the 30-60-90 triangle.
The ratio of the side lengths in a 30-60-90 triangle is 1:√3:2. Since the side length of the rhombus is 16, the length of the shorter leg of the triangle, which is half the length of the diagonal, is 16/2 = 8.
Using the ratio, we can calculate the length of the longer leg of the triangle, which is the length of the diagonal:
Longer leg = (Shorter leg) * √3 = 8 * √3.
So the length of each diagonal is 8 * √3.
Now that we have the length of the diagonals, we can calculate the area of the rhombus:
Area = (diagonal1 * diagonal2) / 2 = (8 * √3) * (8 * √3) / 2 = 64 * 3 / 2 = 192 / 2 = 96.
Therefore, the area of the rhombus is 96.