Answer
(a) CH = $\sqrt 2$
(b) CF = $\sqrt 3$
Work Step by Step
(a)
Each face of a cube is a square and a diagonal divides it into two 45°–45°–90° triangles. Therefore-
In 45°–45°–90° triangle CDH
Diagonal CH = $\sqrt 2 \times $ DH
CH = $\sqrt 2 \times $ 1 = $\sqrt 2$
(b) CHF is a right triangle, right angled at H. Applying Pythagorean theorem-
$CF^{2} $ = $CH^{2} $ + $HF^{2} $
$CF^{2} $ = $(\sqrt 2)^{2} $ + $1^{2} $ [HF is the side of cube]
$CF^{2} $ = $2 + 1 $ = 3
Therefore-
CF = $\sqrt 3$
