In rhombus ABCD, m<ABC = 60 and AB = 4. Find the length of BD and AC, in simplest radical form.

Question

In rhombus ABCD, m<ABC = 60 and AB = 4. Find the length of BD and AC, in

simplest radical form.

Answer 1

Call intersection point of diagonals point E.

We have triangle ABE.

Diagonals of a rhombus bisect pairs of opposite angles, so m<ABE is half of m<ABC = 1/2 of 60 is 30.

Diagonals of a rhombus are perpendicular, so m<BEA is 90.

Thus m<BAE is 180 - 30 - 90 = 60 and we have a 30-60-90 right triangle with hypotenuse AB = 4.

Since AB is the hypotenuse, the side opposite the 30 degree angle (AE) is equal to half the hypotenuse or 2.

and the side opposite the 60 degree angle (BE) is equal to 2* sqrt(3).

Diagonals of a parallelogram bisect one another so AC = 2 * AE = 4 and BD = 2 * BE = 4*sqrt(3).