SOLUTION: QUESTION:In circle O, diameter AB is perpendicular to chord CD at E. If CD=20cm, BE=2cm and AE= 4x-2, find the length of the diameter. Using the intersecting chord theorem I kno

Algebra ->  Circles -> SOLUTION: QUESTION:In circle O, diameter AB is perpendicular to chord CD at E. If CD=20cm, BE=2cm and AE= 4x-2, find the length of the diameter. Using the intersecting chord theorem I kno      Log On


   

Question 992303: QUESTION:In circle O, diameter AB is perpendicular to chord CD at E. If CD=20cm, BE=2cm and AE= 4x-2, find the length of the diameter.
Using the intersecting chord theorem I know how to solve it, however neither values of CE or DE are given and you can't have two variables in an equation so I'm confused. By long shot i thought of using trigonometry but i don't think it's right. Thanks for the help in advance x
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the length of your diameter is 4x-2 + 2 = 4x.
this means your radius is equal to 2x.

if CD is equal to 20, this means that CE and ED are both equal to 10 because a diameter cuts a chord perpendicular to it into 2 equal segments.

here's what you have:

AB = the diameter = 4x.
AO = a radius = 2x.
OB = a radius = 2x.

since EB is equal to 2, then OE must be equal to 2x-2.

CE is equal to 10
ED is equal to 10

OC is equal to 2x because OC is a radius.
OD is equal to 2x because OD is a radius.

you have a right triangle formed that is OCE.
in that triangle:
OC is the hypotenuse which is equal to 2x.
OE is one leg which is equal to 2x-2.
CE is the other leg which is equal to 10.

by pythagorus, OE^2 + CE^2 = OC^2 which becomes:
(2x-2)^2 + 10^2 = (2x)^2
simplify this equation to get:
4x^2 - 8x + 4 + 100 = 4x^2
subtract 4x^2 from both sides of the equation to get:
-8x + 4 + 100 = 0
add 8x to both sides of the equation to get:
4 + 100 = 8x
combine like terms to get:
104 = 8x
solve for x to get:
x = 13.

if x = 13, then:
OC = 2x = 26
OE = 2x-2 = 24
CE = 10

by pythagorus, 10^2 + 24^2 = 26^2
this becomes:
100 + 576 = 676 which becomes:
676 = 676
this confirms the solution is correct.

you are asked to find the length of the diameter.
the length of the diameter is 4x which is equal to 4 * 13 which is equal to 52.

i believe that's your solution.

here's a picture to help you understand what i'm talking about.

$$$

the key to solving is to know the geometric relationship that a chord perpendicular to a diameter is cut in half.

you also needed to determine that the length of the diameters was 4x-2 + 2 which made it equal to 4x.

you also needed to know that the triangle formed had a hypotenuse that was equal to the radius.