A rhombus has an angle measure of 120, and its longer diagonal has a length of 10 inches. Find the area of the rhombus.

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tjbrewer eNotes educator| Certified Educator

A Rhombus is a parallelogram with all congruent sides.  The longer diagonal splits the Rhombus into two obtuse isoceles triangles.  According to Side-Angle-Side Theory these two triangles are congruent.  A triangle's internal angles always add up to 180.  We already know the obtuse angle to be 120, that means the other two angles must add up to 60.  Since the triangle is isoceles, the two angles will be congruent, and therefore each would be half of 60 or 30. 

Now if we draw a vertical line perpendicular to the base from one of the long diagonal corners, we form with the diagonal a right triangle with a Hypotenuse of 10 inches, and an internal angle of 30 opposite the height of the rhombus.  Since the sin of 30 is 1/2 we can then calculate that the Rhombus is 5 inches high. 

To find the base we observe that the vertical we drew also forms another right triangle with one side of the rhombus as it's hypotenuse, and an angle opposite the height of 60.  The Sin of 60 is about 866/1000.  So we have the calculation `5/b~~866/1000` Cross multiply and we get `866b~~5000` So that gives us an approximate length of b=5.77 inches. 

Now that we know the base and height of the rhombus, we can use the formula for the area of a Parallelogram (since a Rhombus is a parallelogram) to calculate it's area.  A=bh.  b=5.77, h=5  A=5.77x5=28.85.  The area of the Rhombus is 28.85 Square Inches.