Circle Γ with center O has diameter AB=192. C is a point outside of Γ, such that D is the foot of the perpendicular from C to AB and D lies on the line segment OB. From C, a tangent to Γ is drawn, touching Γ at E, where the foot of the perpendicular from E to AB lies within AD. CD intersects EB at F. If CF=110, what is the length of OC?

145 wrong answer..

156

234 Make a line segment OE. Since OE=OB, angle OEB = angle OBE. This implies that angle FEC = 90 - angle OEB = 90 - angle OBE = angle DFB = angle EFC.

Thus angle FEC = angle EFC. Thus EC = CF = 110.

Now in right angle triangle CEO, CE = 110, OE = 96. By Pythagoras's theorem

OC^2 = OE^2 + EC^2 = 110^2 + 96^2 = 21316. This implies OC = 146

the correct ans is 146.....and say Thanks.....

To find the length of OC, we need to use some properties of circles and triangle similarity.

First, we know that OC is a radius of circle Γ since O is the center. Let's call the length of OC as x.

Next, let's analyze the given information. We have CD = 110 and CF = 110. Since D lies on the line segment OB, we also have DB = 192 - CD = 192 - 110 = 82.

Using these lengths, we can start by finding the length of EF. Since EF is a tangent to circle Γ, it is perpendicular to OE at point E. This means that triangle ECO is similar to triangle EDF.

We can set up a similarity ratio using the corresponding sides of these triangles:

EC/ED = CO/DF

Substituting the known values, we have:

x/110 = x + 82/192

To solve for x, let's cross-multiply:

192x = 110(x + 82)

Distributing and rearranging the terms:

192x = 110x + 110(82)
192x = 110x + 9020
82x = 9020
x = 9020/82
x ≈ 109.756

Therefore, the length of OC is approximately 109.756 units.