Two sides of a rhombus form a 120º angle. The length of each side is 6 in. Explain how to find the area of the rhombus, and then calculate the area. Leave your answer in simplest radical form.
i have the same question in my class too so thank you
If X = 9 cm and Z = 15 cm, what is the length of Y
To find the area of a rhombus, you can use the formula:
Area = (diagonal1 * diagonal2) / 2
In this case, we have a rhombus with two sides forming a 120º angle. Since opposite angles of a rhombus are congruent, the other two sides will also form a 120º angle. We can use this angle to find the diagonals of the rhombus.
Since the diagonals of a rhombus bisect each other at right angles, the angle between the diagonals is 90º. Let's denote the intersection point of the diagonals as point O.
Now, draw a line segment from point O to one of the sides. This will divide the rhombus into two congruent right-angled triangles.
We can use the trigonometric properties of a right-angled triangle to find the lengths of the diagonals. In this case, we have a 30-60-90 triangle since one angle is 90º and another angle (formed by the diagonal and one side) is 30º.
In a 30-60-90 triangle, the ratio of the side lengths is 1:√3:2. Since the length of one side of the rhombus is given as 6 inches, the length of the shorter diagonal (2) will be twice this length, which is 12 inches.
To find the length of the longer diagonal, we can use the Pythagorean theorem:
(longer diagonal)^2 = (shorter diagonal)^2 + (side length)^2
(longer diagonal)^2 = 12^2 + 6^2
(longer diagonal)^2 = 144 + 36
(longer diagonal)^2 = 180
(longer diagonal) = √180 = 6√5
Now that we have the lengths of the diagonals, we can substitute them into the area formula:
Area = (diagonal1 * diagonal2) / 2
Area = (12 * 6√5) / 2
Area = 72√5 / 2
Area = 36√5
Therefore, the area of the rhombus is 36√5 square inches.
If two angles are 120, then the other two must total 360-2*120 = 120
So each other angle is 60.
When you draw diagonals in a rhombus, they form right angles at their intersection.
So, what we have after we draw in the diagonals, is 4 triangles. Each triangle is 30-60-90. And the hypotenuse is 6.
Using the properties of a 30-60-90 you can get the length of the other two legs.
Once you have that info, finding the area is easy A=bh/2
Since you have 4 congruent triangles, the total area is 4 times the area of a single triangle