A cube has unit length sides. What is the length of a line joining a vertex to the midpoint of oneof the opposite faces (the dashed line in the diagram below)?

Answer:

8) B

Let the dashed line be called L.
Let length of each side = 1

We first need to calculate the length of half of the diagonal of one of the faces.
Let diagonal be = D

Using Pythagoras,
D² = 1² + 1²
D² = 1 + 1 = 2
D = √2

Half of D = √2
                    2

Imagine the dashed line forming a triangle with the vertex just below.

Using Pythagoras,
L² = 1² + (√2 / 2)²
L² = 1 + (2 / 4) = 1 + (1/2) = 1.5
L = √1.5 = √³⁄₂

So dashed line = √³⁄₂