Answer:
33 mm
Explanation:
Given a triangle with sides 48 mm and 55 mm, and angles x and w such that cos(x) = 4/5 and cos(w) = 1/2, you want the length of the third side (r) opposite angle x.
Law of cosines
For sides a, b, c and angle C opposite side c, the Law of Cosines gives the relation ...
c² = a² +b² -2ab·cos(C)
Application
Filling in the given values, we have ...
r² = 48² +55² -2·48·55·(4/5) = 1105
r = √1105 ≈ 33 . . . . millimeters
The length r is about 33 mm.
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Additional comment
Side r is irrational, so the triangle cannot have exactly the measures shown. Using other measures as true, the value of cos(w) is about 0.4994, not 1/2.
Taking cos(w) as true, side r is found using the law of cosines as 33.43717. Taking cos(x) as true, side r is found using the law of cosines as 32.24154. Taking the angle measures and 48 mm to be true, side r is found using the law of sines as 33.25538. All round to 33 mm.
The attachment shows the solution using cos(w)=1/2, or w=60°.