Question

Here is the question...

Answer 1

Cos60 degrees=1/2, so AK/AB=1/2. Since AK=KD, AK=1/2AD=1/2AB. Therefore, AB=AD. This is a rhombus, with four equal sides. Triangle ABK is congruent to triangle DBK (SAS), since AK=KD, angle AKB=angle BKD=90, and BK=BK. Therefore, BD=AB. The sum of four side lengths is 24. Each side length is equal to 24/4=6. BD=6.

Answer 2

The correct answer is:

BD = 6.

Explanation:

Since this is a parallelogram, we know that opposite sides are congruent; this means that BC = AD.

We have that ∠A = 60°.
cos 60° = 1/2.
The ratio for cosine is adjacent/hypotenuse; the side adjacent to ∠A is AK, and the hypotenuse of the triangle containing ∠A is AB; this means that

AK/AB = 1/2.

Multiplying both sides by AB, we have
AK = 1/2(AB).

Since AK = KD, this means that AK = 1/2(AD).

This gives us 1/2(AD) = 1/2(AB); this means that AD = AB.

We know that AD = BC and AB = CD; we know have AD = AB = BC = CD.  All 4 sides are congruent, which makes this a rhombus.

Since all 4 sides are congruent and the perimeter is 24, we can divide to find the length of each side:
24/4 = 6.

This means that BD = 6.