$$\text{In}△ABC,\mathrm{AB}=\mathrm{AC}=12\text{cm and}\mathrm{ D }\text{is a point on side}\mathrm{ BC }\text{such that}\mathrm{ AD}=8\text{cm. If}\mathrm{AD}\text{is extended to a point }E\text{ such that}\angle \mathrm{ACB}=\angle \mathrm{AEB},$$

$$\text{then the length, in cm, of}\mathrm{ AE}\text{ is}$$

The correct option is 18.

To find the length of the segment $AE$, we can use the properties of similar triangles. Let's break down the proof and calculations step-by-step:

Step 1: Understand the given information
We are given an isosceles triangle $△ABC$ where:
$$AB=AC=12\text{cm}$$
Since $AB=AC$, the angles opposite to these sides must also be equal. Therefore, the base angles are equal:
$$\angle ABC=\angle ACB$$
Since point $D$ lies on the line segment $BC$, we can also write:
$$\angle ABD=\angle ACB$$

Step 2: Establish the relationship with the extended point E
We are given that the line segment $AD$ is extended to point $E$ such that:
$$\angle ACB=\angle AEB$$
Combining this with our equation from Step 1, we get:
$$\angle ABD=\angle AEB$$

Step 3: Identify similar triangles
Now let us compare triangle $△ABD$ and triangle $△AEB$:
1. $\angle BAD=\angle EAB$ (since they refer to the exact same shared angle $\angle BAE$).
2. $\angle ABD=\angle AEB$ (established in Step 2).
By the Angle-Angle (AA) similarity criterion, the two triangles are similar:
$$△ABD\sim △AEB$$

Step 4: Set up the ratio and calculate AE
Since the triangles are similar, the ratios of their corresponding sides must be equal:
$$\frac{AB}{AE}=\frac{AD}{AB}$$
Cross-multiplying the terms gives us:
$$AE\cdot AD=A{B}^2$$
We are given that $AB=12\text{cm}$ and $AD=8\text{cm}$. Substituting these values into the equation:
$$AE\cdot 8={12}^2$$
$$AE\cdot 8=144$$
Solve for $AE$:
$$AE=\frac{144}{8}=18\text{cm}$$