The company Real Steel has two manufacturing plants located 12 km away from each other. They also have an underground pipeline that provides water from the lake used for cooling the metal. Real Steel wants the cheapest possible pipeline so they decide to run pipes from each plant that meet at the middle (as shown below). Furthermore, the distance from the plants to the lake is 10 km. What distance between where the lake and where the pipelines meet minimizes the total length of pipes used? Additionally, show it is a minimum.