In the following figure,

$$$$ CD = 5 cm , BE = 10 cm , AE = 12 cm,

∠ DAB = ∠ DCB , and  ∠ DAE = ∠ DBC = 90°

Points AFCD create a rhombus.

The correct answer is 3.

Here is the step-by-step derivation to find the length of segment BD:

1. Understand the properties of the rhombus:
We are given that points A, F, C, and D create a rhombus. A key property of a rhombus is that all of its sides are equal in length.
Since we are given:
$\mathrm{CD}=5\mathrm{cm}$
It follows that:
$\mathrm{AD}=\mathrm{CD}=5\mathrm{cm}$

2. Analyze the right-angled triangle DAE:
We are given that:
$\angle \mathrm{DAE}=90°$
This means triangle DAE is a right-angled triangle with the hypotenuse DE. We are given the length of side AE:
$\mathrm{AE}=12\mathrm{cm}$

Applying the Pythagorean theorem to triangle DAE:
${\mathrm{DE}}^2={\mathrm{AD}}^2+{\mathrm{AE}}^2$

Substitute the values of AD and AE into the equation:
${\mathrm{DE}}^2=5^2+{12}^2$
${\mathrm{DE}}^2=25+144$
${\mathrm{DE}}^2=169$

Taking the square root of both sides:
$\mathrm{DE}=\sqrt{169}=13\mathrm{cm}$

3. Calculate the length of BD:
From the given figure, the points D, B, F, and E lie along the collinear vertical diagonal axis of the rhombus. Therefore, the segment DE is composed of segments BD and BE:
$\mathrm{DE}=\mathrm{BD}+\mathrm{BE}$

We are given:
$\mathrm{BE}=10\mathrm{cm}$

Substitute the known lengths of DE and BE into the relation:
$13=\mathrm{BD}+10$
$\mathrm{BD}=13-10$
$\mathrm{BD}=3\mathrm{cm}$