Arhombus abcd has ab = 10 and m∠a = 60°. find the lengths of the diagonals of abcd.

Since it is a rhombus, all sides will measure 10 units. Since the diagonals of a rhombus are angle bisectors, then diagonal AC bisects angle A. Also, the diagonals of a rhombus are perpendicular bisectors of each other, so 4 congruent right triangles are formed. Consider one of these right triangle, we have 30°, with hypotenuse 10. We can easily solve for the remaining 2 sides of the triangle. Let E be the intersection of the two diagonals.
cos(30°) = AE/10
AE = 5√3
AC = 2AE = 10√3

sin(30°) = BE/10 = 5
BD = 2BE = 10

The lengths of the diagonals are 10√3 units and 10 units.

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