What is the value of x? Triangle V T K with segment T Y such that Y is on segment V K, between V and K. Angle V T Y is congruent to angle Y T K. V T equals 57 millimeters, V K equals x, Y K equals 68 millimeters, and T K equals 129.2 millimeters.

Answer: 98 millimeters

Explanation:
 
Since angle VTY is congruent to angle VTK, segment TY bisects angle VTK. Since Y is on segment VK, between V and K, we can use the Angle Bisector Theorem, which states that

[tex] \frac{VY}{YK} = \frac{VT}{TK} [/tex]    (1)

Since x= VK = VY + YK, we need to obtain VY since YK = 68.

VY is obtained by multiplying the denominator YK on both sides of equation (1). So, 

[tex]VY = \frac{(VY)(VT)}{TK} = \frac{(68)(57)}{129.2} \newline VY = 30[/tex]

Hence,

x = VK = VY + YK
x = 30 + 68
x = 98 millimeters