Answer:
![x=4\sqrt{6}\ units](https://tex.z-dn.net/?f=x%3D4%5Csqrt%7B6%7D%5C%20units)
![y=4\sqrt{2}\ units](https://tex.z-dn.net/?f=y%3D4%5Csqrt%7B2%7D%5C%20units)
![z=4\sqrt{3}\ units](https://tex.z-dn.net/?f=z%3D4%5Csqrt%7B3%7D%5C%20units)
Step-by-step explanation:
The picture of the question in the attached figure
step 1
In the right triangle ABD
Applying the Pythagorean Theorem
![x^2=y^2+(12-4)^2](https://tex.z-dn.net/?f=x%5E2%3Dy%5E2%2B%2812-4%29%5E2)
![x^2=y^2+64](https://tex.z-dn.net/?f=x%5E2%3Dy%5E2%2B64)
----> equation A
step 2
In the right triangle BDC
Applying the Pythagorean Theorem
![z^2=y^2+4^2](https://tex.z-dn.net/?f=z%5E2%3Dy%5E2%2B4%5E2)
![z^2=y^2+16](https://tex.z-dn.net/?f=z%5E2%3Dy%5E2%2B16)
----> equation B
step 3
In the right triangle ABC
Applying the Pythagorean Theorem
![12^2=x^2+z^2](https://tex.z-dn.net/?f=12%5E2%3Dx%5E2%2Bz%5E2)
----> equation C
step 4
Equate equation A and equation B
![x^2-64=z^2-16](https://tex.z-dn.net/?f=x%5E2-64%3Dz%5E2-16)
-----> equation D
step 5
substitute equation D in equation C
![144=z^2+48+z^2](https://tex.z-dn.net/?f=144%3Dz%5E2%2B48%2Bz%5E2)
solve for z
![2z^2=144-48](https://tex.z-dn.net/?f=2z%5E2%3D144-48)
![2z^2=96](https://tex.z-dn.net/?f=2z%5E2%3D96)
![z^2=48](https://tex.z-dn.net/?f=z%5E2%3D48)
![z=\sqrt{48}\ units](https://tex.z-dn.net/?f=z%3D%5Csqrt%7B48%7D%5C%20units)
simplify
![z=4\sqrt{3}\ units](https://tex.z-dn.net/?f=z%3D4%5Csqrt%7B3%7D%5C%20units)
Find the value of x
![x^2=48+48=96](https://tex.z-dn.net/?f=x%5E2%3D48%2B48%3D96)
![x=\sqrt{96}\ units](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B96%7D%5C%20units)
![x=4\sqrt{6}\ units](https://tex.z-dn.net/?f=x%3D4%5Csqrt%7B6%7D%5C%20units)
Find the value of y
![y^2=z^2-16](https://tex.z-dn.net/?f=y%5E2%3Dz%5E2-16)
![y^2=48-16](https://tex.z-dn.net/?f=y%5E2%3D48-16)
![y^2=32](https://tex.z-dn.net/?f=y%5E2%3D32)
![y=\sqrt{32}\ units](https://tex.z-dn.net/?f=y%3D%5Csqrt%7B32%7D%5C%20units)
![y=4\sqrt{2}\ units](https://tex.z-dn.net/?f=y%3D4%5Csqrt%7B2%7D%5C%20units)