Let A and B be points on a circle of radius 2 centered at O, such that angle AOB = 90. C is the midpoint of OB, and AC extended meets the circle at D. Find the length CD.
+128460 See the following image :
Let O = (0,0)
Let A = (2,0)
Let B = (0,2)
Let E = (-2,0)
C =(0,1)
Note that BE is the diameter = 4
And CE = BE - BC = 4 - 1 = 3
Since AOB is right with OA = 2 and OC = 1
Then, by the Pythagorean Theorem AC =sqrt ( OA^2 + OC^2) = sqrt ( 2^2 + 1^2) = sqrt (5)
By the intersecting chord theorem we have that
BC * CE = AC * CD
1 * 3 = sqrt (5) * CD
3 = sqrt (5) * CD
CD = 3/sqrt (5) ≈ 1.34
