Tangent and Circle ProblemA large circle has centre C and diameter AB. A smaller circle has centre D and diameter BC. Chord AE is tangent ot the smaller circle. If AB = 18 cm, what is the exact...

Tangent and Circle Problem

A large circle has centre C and diameter AB. A smaller circle has centre D and diameter BC. Chord AE is tangent ot the smaller circle. If AB = 18 cm, what is the exact length of AE in radical form?

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beckden | High School Teacher | (Level 1) Educator

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The triangle AED is a right triangle with angle AED is a right angle.

So AE^2+DE^2=AD^2

DE=1/2BC and BC=1/2AB so DE=1/4AB = 18/4 cm.

CD=1/2BC  and BC=1/2AB so CD=1/4AB

And

AD=AC+CD=1/2AB+1/2BC=1/2AB+1/2(1/2AB)=3/4AB

So `AE^2=AD^2-DE^2 = (3/4)^2(AB)^2-(1/4)^2(AB)^2=(9/16-1/16)(AB)^2`

AE^2=1/2(AB)^2

So

`AE = sqrt(2)/2AB=9sqrt(2)` cm

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