The three vectors form an isosceles triangle. The dashed line bisects it into two right triangles. To get the equation, consider one of the right triangles.

If you take the diagram at the top left of page 26, and fill in the hypotenuse which is ##\vec A (t + \Delta t)##, you get a similar diagram, but with the right angle in a different location. The discrepancy disappears in the limit as ##\Delta \theta \rightarrow 0## and ##\Delta \vec A \rightarrow 0##.

Ok that makes it clear. But it raises another question on why the vector ##\Delta A## is not perpendicular to ##A##? Whereas the book says it is necessary for it to be so because ##A## is not changing in magnitude?

In the limit as Δt goes to zero (which is what you need to do in order to have the instantaneous derivative), ΔA becomes perpendicular to both A(t) and A(t+Δt). And of course A(t) and A(t+Δt) become equal to each other.