1. The problem is to understand and simplify the expression $$D^2 = (4k)^2 + (3k)^2$$ and explain where the variable $k$ comes from. 2. This expression looks like the Pythagorean theorem formula for the length of the hypotenuse $D$ of a right triangle with legs $4k$ and $3k$. 3. The variable $k$ is a scaling factor that multiplies the sides of a basic 3-4-5 right triangle. The original triangle has sides 3, 4, and 5. When scaled by $k$, the sides become $3k$, $4k$, and the hypotenuse $D$. 4. Using the Pythagorean theorem: $$D^2 = (4k)^2 + (3k)^2$$ 5. Calculate each term: $$D^2 = 16k^2 + 9k^2$$ 6. Combine like terms: $$D^2 = 25k^2$$ 7. Take the square root of both sides: $$D = \sqrt{25k^2}$$ 8. Simplify the square root: $$D = 5k$$ 9. So, the hypotenuse $D$ is $5k$, which confirms the triangle is a scaled version of the 3-4-5 triangle by the factor $k$. This explains that $k$ is the scale factor that stretches or shrinks the original triangle while keeping the same shape.