secant AB intersects circle O at D, secant AOC intersects circle O at E, AE=4,AB=12, and DB=6. What is the length of OC?

Question

Choices are:

a. 9

b. 7

c. 4.5

d. 14

 

I already tried x² =(12)(6) and I got 6 square root 2 and also attempted 4x=(12)(6), which=18.

Answer 1

Caleb, as I read this your second attempt 4x=(12)(6) gives x=18, but x≠OC, x=AC.

 

The secant formula is that the product of the full length of the secant from A to where it exits the Circle at B times the length of the secant from A to where it enters the circle at D is equal to the product of the full length of the secant from A to where it exits the circle at C times the length of the secant from A to where it enters the Circle at E.  Or algebraically, (AB)(AD)=(AC)(AE), but we are given AE, AB and DB. 

 

You need to use other geometry principles to calculate what you need.  And we know that AD + DB = AB, and we know that AE + EC = AC.  Finally, we know that EOC must be a diameter and OC a radius of the circle, so OC = 1/2 EC.

 

(12)(AD)=(AC)(4)

Since AD+DB=AB, AB-DB=AD, so 12-6=AD=6

(12)(6)=(AC)(4), so AC=18 [You did all the work up to this point]

AOC=AE+EOC, so 18=4 + EOC, so EOC = 14.

Since EOC is a diameter, OC is a radius and thus equal to half of EC or 7.