In a parallelogram, the acute angle is 60 degrees, the height of the parallelogram drawn from the top of the obtuse

In a parallelogram, the acute angle is 60 degrees, the height of the parallelogram drawn from the top of the obtuse angle divides the sides of the parallelogram in half. Find the smaller diagonal of the parallelogram if the perimeter is 24 cm

1. Let’s designate the vertices of the parallelogram ABCD, the height BH.

2. Angle ABH: 180 ° – 90 ° – 60 ° = 30 °.

2. Leg AH = AB / 2, as it is opposite an angle of 30 °.

3. Side АD = АН + DH = 2АН, since АН = DH by condition.

4. Substitute in this expression AH = AB / 2:

AD = 2AB / 2 = AB.

5. We calculate the length AB:

2AB + 2AB = 24;

AB = 6 cm.

6. Triangles ABH and BDH are equal, since legs AH and DH are equal, and BH is a common leg.

AB = BD = 6 cm, so they are against equal angles.

Answer: the smaller diagonal BD of the parallelogram is 6 cm.