In XYZ , XZ=8 , YZ=5 , and XY=7 .

Since you know the lengths of the sides, you can use Heron's formula:

     A  =  sqrt[ s · (s - a) · (s - b) · (s - c) ]

where  a,  b,  and  c  are the sides 

and  s  is the semiperimeter:  s  =  (a + b + c) / 2.

First:  find the semiperimeter:  a  =  (8 + 5 + 7) / 2  =  10

Then:  find the area:  A  =  sqrt[ 10 · (10 - 8) · (10 - 5) · (10 - 7) ]  =  sqrt[ 300 ]

$\cos \angle YZX = \dfrac{5^2 + 8^2 - 7^2}{2(5)(8)} = \dfrac12$

Therefore $\angle YZX = 60^\circ$.

Area = $\dfrac12 \cdot 5 \cdot 8 \sin 60^\circ = 10 \sqrt 3$