In a circle, the distance from a chord of length 12 units to the midpoint of its minor arc is 4 units. What is the radius of the circle?

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Answer 1 · 2018-06-26T10:29:20

$Radius = \frac{13}{2}$ $color(blue)("Radius"=13/2)$

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Using diagram:

We are given:

$A B = 12$ $AB=12$

$C D = 4$ $CD=4$

Note that the midpoint of the arc $ˆ A C B$ $hat(ACB)$ is also the midpoint of $A B$ $AB$

$:.$

$AD=DB=1/2(AB)=6$

If we extend $CD$ through the centre to $E$ , we form the chord $CE$ .

We now have two intersecting chords:

By the intersecting chords theorem:

$ADxxDB=CDxxDE$

$:.$

$6xx6=4xxDE$

$DE=36/4=9$

Diameter

$CE = DE+CD$

$\ \ \ \ \ \ 9+4=13$

$"Radius"=(CE)/2=13/2$