Explore the concept of half-slope in relation to planar reflections and rotations, and its connection to the rational parametrization of the unit circle. Learn how to recast the algebraic understanding of composition of grounded isometries using half-slopes, including the circle sum of half-slopes and formulas for higher circle sums. Discover the full group of Euclidean isometries by introducing translations as the third type of isometry. This 21-minute video provides an in-depth look at these mathematical concepts, offering a fresh perspective on understanding and working with isometries in the Euclidean plane.