# Why do we scale by 10/ sqrt 2?

• Module 2 Week 1 Day 2 Challenge Part 3

Sometimes my brain needs a second to recover from this stuff. Why do we scale by 10/ sqrt 2?

Let's consider a triangle that will be similar to our isosceles right triangles and whose sides are given. One such triangle will be a $$1-1-\sqrt{2}$$ right triangle. The ratios of these similar triangles will be:
$$\frac{{\color{orange}\text{\bf{h}}-\text{\bf{h}}-\bf{5}\text{ right triangle}}}{1-1-\sqrt{2}\text{ right triangle}}={\color{orange}\boxed{\frac{5}{\sqrt{2}}}}=\frac{\color{orange}\text{\bf{h}}}{1}=\color{orange}\text{\bf{h}}$$
$$\frac{{\color{blue}\text{\bf{h}}-\text{\bf{h}}-\bf{13}\text{ right triangle}}}{1-1-\sqrt{2}\text{ right triangle}}={\color{blue}\boxed{\frac{13}{\sqrt{2}}}}=\frac{\color{blue}\text{\bf{h}}}{1}=\color{blue}\text{\bf{h}}$$
$$\frac{{\color{green}\text{\bf{h}}-\text{\bf{h}}-\bf{25}\text{ right triangle}}}{1-1-\sqrt{2}\text{ right triangle}}={\color{green}\boxed{\frac{25}{\sqrt{2}}}}=\frac{\color{green}\text{\bf{h}}}{1}=\color{green}\text{\bf{h}}$$
$$\frac{{\color{purple}\text{\bf{h}}-\text{\bf{h}}-\bf{10}\text{ right triangle}}}{1-1-\sqrt{2}\text{ right triangle}}={\color{purple}\boxed{\frac{10}{\sqrt{2}}}}=\frac{\color{purple}\text{\bf{h}}}{1}=\color{purple}\text{\bf{h}}$$
So, in order to find $$\color{purple}\text{h}$$ we just need to find the scale factor of our triangle and its similar $$1-1-\sqrt{2}$$ right triangle.