see the attached figure to better understand the problem
we have that
[tex]cos(A)=\frac{1}{5} \\ AB=12\ units\\ CD=2\ units[/tex]
Step 1
Find the value of AC
we know that
in the right triangle ABC
[tex]cos (A)=(AC/AB)\\AC=AB*cos(A)[/tex]
substitute the values in the formula
[tex]AC=12*(1/5)\\ AC=2.4\ units[/tex]
Step 2
Find the value of BC
we know that
in the right triangle ABC
Applying the Pythagorean Theorem
[tex]AB^{2} =AC^{2}+BC^{2}\\ BC^{2}=AB^{2} -AC^{2}[/tex]
substitute the values
[tex]BC^{2}=12^{2} -2.4^{2}\\BC^{2}= 138.24\\ BC=11.76\ units[/tex]
Step 3
Find the value of BD
we know that
in the right triangle BCD
Applying the Pythagorean Theorem
[tex]BD^{2} =DC^{2}+BC^{2}[/tex]
substitute the values
[tex]BD^{2} =2^{2}+11.76^{2}[/tex]
[tex]BD=11.93\ units[/tex]
therefore
the answer is
the length of BD is 11.93 units
