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geometry

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Whats the area of a rhombus thats angles are 120, 120, 60, and 60 who's sides all equal 1.

Jan 31, 2021

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First and foremost, I would draw a diagram. This is a good start with any geometry problem. I have constructed a mock version of a sketch and presented it below. This way, both of us can visualize the problem together.

To be honest, I either never learned the formula for the area of a rhombus, or I forgot the formula. Either way, I will have to resort to some prior knowledge that I have in order to figure out the area of this rhombus. By definition, rhombuses are special parallelograms, so the area formula for a parallelogram will hold true for a rhombus.

$$A_{parallelogram}=bh$$ where $$b$$ is the length of one of the bases and $$h$$ is the perpendicular height of the parallelogram. I have labeled these lengths in the diagram above as well.

$$b=AB=1$$ as the length of any side of a rhombus is 1, as given by the original problem.

Finding $$h$$ requires a little bit more work, but not too much either. $$\triangle EDA$$ is a right triangle where $$DA=1$$ because that is a length of a side length of a rhombus. $$m\angle EAB = 120^\circ$$, which is, once again, given information. It is also possible to find the measures of the angles of the triangles, which will also reveal the length of the side of $$\triangle EDA$$

$$m\angle EAD + m\angle DAB = m\angle EAB\\ m\angle EAD + 90^\circ= 120^\circ\\ m\angle EAD= 30^\circ$$

$$m\angle DEA + m\angle EAD + m\angle ADE = 180^\circ \\ m\angle DEA + 30^\circ + 90^\circ = 180^\circ \\ m\angle DEA = 60^\circ$$

Now, we have determined that $$\triangle EDA$$ is a 30-60-90 triangle, which also means that its side lengths have a consistent ratio of $$1:\sqrt{3}:2$$. Let's set up a proportion then to find the length of the perpendicular height of the triangle.

$$\frac{\sqrt{3}}{2}=\frac{h}{1}\\ h=\frac{\sqrt{3}}{2}$$

Now that we have both b and h, we can now solve for the area of this mystery rhombus.

$$A_{\text{rhombus}}=bh\\ A_{\text{rhombus}}=1*\frac{\sqrt{3}}{2}\\ A_{\text{rhombus}}=\frac{\sqrt{3}}{2}$$

I hope this explanation helped you!

Jan 31, 2021