uncertain point in textbook's solution of a "distance from point to line" problem

Question

We have a cube with each side equaling $1$ unit of length. We need to determine the distance from $B$ to the line $A_1C_1$

In my calculation, each side of the yellowish-shaded triangle equals $\sqrt2$. Then we just solve

$$d=\sqrt{2-\frac24}=\frac{\sqrt3}{2}$$

But the textbook says:

Why does the author uses the additional factor $BA_1$?

Sorry if the question is too simple, I've been remiss in my studies, am now trying to catch up. (0:

Answer 1

$2-\frac24=\frac32$, not $\frac34$. With that corrected, your answer from the pythagorean theorem is equal to the book's.

The book is using a different method, noting that the $BA_1C_1$ is equilateral, and is scaling up the known height of an equilateral triangle of side length 1.

Answer 2

Your method is correct, however in your answer you should have $\sqrt2$ in the denominator, not just a $2$ When you rationalize your denom it comes out ok

Answer 3

Observe: $$\sqrt{2-\frac{1}{2}}=\frac{\sqrt{3}}{\sqrt{2}}=\frac{\sqrt{6}}{2}$$