Answer:
x = 8
Explanation:
You want to know the value of x in this similar triangle geometry.
Similar triangles
When a segment is drawn in a triangle parallel to the base, it divides the sides proportionally. Every segment on the left has the same ratio to the corresponding segment on the right:
x/6 = 4/3
This equation is solved in the usual way. Multiply by the inverse of the coefficient of x:
6(x/6) = 6(4/3)
x = 8
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Additional comment
Every proportion can be written several ways. Another way to write this one is ...
x/4 = 6/3
x = 4(2) = 8 . . . . . multiply by 4
If you need to, you can prove the triangles are similar by considering corresponding angles. ∠A≅∠A, ∠APW≅∠AVZ, so the triangles are similar by the AA postulate. (Corresponding angles where a transversal crosses parallel lines are congruent.)
Perhaps not so obvious is the relationship between segments of the long side and segments of the short side. The similarity relation tells us ...
AV/AP = AZ/AW
(4+x)/x = (3+6)/6
4/x +1 = 3/6 +1 . . . . . do the division
4/x = 3/6 . . . . . . . . subtract 1
x/4 = 6/3 . . . . . . invert both sides . . . compare to above