Answer:
[tex]x=\sqrt{41}\ units[/tex]
Step-by-step explanation:
see the attached figure with letters to better understand the problem
we know that
In a right triangle the Pythagoras theorem formula is equal to
[tex]c^{2} =a^{2} +b^{2}[/tex]
where
c is the hypotenuse
a,b are the legs of triangle
step 1
In the right triangle ABC find BC
[tex]AB^{2} =AC^{2} +BC^{2}[/tex]
substitute the values
[tex]52^{2} =48^{2} +BC^{2}[/tex]
[tex]BC^{2}=52^{2}-48^{2}[/tex]
[tex]BC^{2}=400[/tex]
[tex]BC=20\ units[/tex]
step 2
In the right triangle CBF find CF
[tex]BC^{2} =BF^{2} +CF^{2}[/tex]
substitute the values
[tex]20^{2} =12^{2} +CF^{2}[/tex]
[tex]CF^{2}=20^{2}-12^{2}[/tex]
[tex]CF^{2}=256[/tex]
[tex]CF=16\ units[/tex]
step 3
In the right triangle DEF find EF
[tex]DF^{2} =ED^{2} +EF^{2}[/tex]
substitute the values
[tex]13^{2} =5^{2} +EF^{2}[/tex]
[tex]EF^{2}=13^{2}-5^{2}[/tex]
[tex]EF^{2}=144[/tex]
[tex]EF=12\ units[/tex]
step 6
Find the measure CE
[tex]CE=CF-EF[/tex]
substitute
[tex]CE=16-12=4\ units[/tex]
step 7
In the right triangle CDE find the value of x
[tex]x^{2} =CE^{2} +ED^{2}[/tex]
substitute the values
[tex]x^{2} =4^{2} +5^{2}[/tex]
[tex]x^{2}=41[/tex]
[tex]x=\sqrt{41}\ units[/tex]
