If line segment ab is a diameter and is at right angles to line segment cd, what is the ratio of cd to de?

The answer to this is 2.

There are a number of proofs to this. Here, we use Euclidean geometry with trigonometry. If we let the center of the circle to be O.
Then, we have the following equations for the angles
CEO = OED = 90

Since, CO = OD because they're radii of the circle, then
ΔCOD is an isosceles triangle and
OCE = ODE

CE + DE = CD
dividing the whole equation by DE
CE/DE + 1 = CD/DE

Using trigonometric functions:
CE = OC cos OCE and
DE = OD cos ODE

Substituting.
OC cos OCE / OD cos ODE + 1 = CD/DE

Since, OCE = ODE,
cos OCE = cos ODE

The equation would be reduced to:
1 + 1 = CD/DE
CD/DE =2