Choice B is correct. It's given that in $△ A B C$ , $∠ B$ is a right angle. Therefore, $△ A B C$ is a right triangle, and $\bar{A C}$ is the hypotenuse of the triangle. It's also given that $\cos A = \frac{3}{5}$ . Since the cosine of an acute angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, the ratio of the length of $\bar{A B}$ to the length of $\bar{A C}$ is $3$ to $5$ . It follows that the length of $\bar{A B}$ can be represented as $3 a$ and the length of $\bar{A C}$ can be represented as $5 a$ , where $a$ is a constant. The Pythagorean theorem states that the sum of the squares of the length of the legs of a right triangle is equal to the square of the length of its hypotenuse, so it follows that $A B^{2} + B C^{2} = A C^{2}$ . Substituting $3 a$ for $A B$ and $5 a$ for $A C$ in this equation yields ${(3 a)}^{2} + B C^{2} = {(5 a)}^{2}$ , or $9 a^{2} + B C^{2} = 25 a^{2}$ . Subtracting $9 a squared$ from both sides of this equation yields $upper B upper C squared equals 16 a squared$ , or $upper B upper C equals 4 a$ . It follows that the ratio of the length of $\bar{A B}$ to the length of $\bar{B C}$ is $3$ to $4$ . Let $x$ represent the length, in millimeters, of $\bar{A B}$ . It's given that the length of $\bar{B C}$ is $136$ millimeters. Since the ratio of the length of $\bar{A B}$ to the length of $\bar{B C}$ is $3$ to $4$ , $\frac{x}{136} = \frac{3}{4}$ . Multiplying both sides of this equation by $136$ yields $x = \frac{3 (136)}{4}$ , or $x equals 102$ . Therefore, the length of $\bar{A B}$ is $102$ millimeters.

Choice A is incorrect. This is the scale factor by which the $3$ to $4$ to $5$ ratio is multiplied that results in the side lengths of $△ A B C$ .

Choice C is incorrect. This is the length of $\bar{B C}$ , not the length of $\bar{A B}$ .

Choice D is incorrect. This is the length of $\bar{A C}$ , not the length of $\bar{A B}$ .