Functions Overloading in interface and classes - how to?

When you declare the function in the class you need to decorate it with the overloads:

getDist(): string;
getDist(x: number): any;
getDist(x?: number): any {
    // your code
 }

This answer describes how to implement method overloading in TypeScript, and it's not pretty:

interface IPoint {
    getDist(): string;
    getDist(x: number): any;
}

class Point implements IPoint {
    // Constructor
    constructor (public x: number, public y: number) { }

    pointMethod() { }

    getDist(x?: number) {
         if (x && typeof x == "number") {
             return 'foo';
         } else {
             return 'bar';
         }
    }
}

N.B. with the particular combination of declared return types in the interface, you are limited to returning strings from getDist.