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# I'm stuck on this geometry question

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The line y=(3x+20)/4 intersects a circle centered at the origin at A and B. We know the length of chord AB is 20. Find the area of the circle.

May 8, 2021

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I'll admit that  this one  baffled  me  for a moment, too.....but

(1)  First find  the distance between  the origin and the line using the formula  in the other problem

The  equation of the line  can  be  re-wriiten as

4y = 3x + 20

3x - 4y +  20  =  0

l  3(0) - 4(0)   +  20 l  / sqrt  ( 3^2 + 4^2)  =

20  / sqrt 25

20 / 5   =   4

Now....if  we  drew a perpendicular  to  the line from the origin  , this would  meet the  chord at its midpoint

And  the  length of the perpendicular   and 1/2  the  chord length form legs of a right triangle with  the  radius of the circle as  the hypotenuse

Using  the Pythagorean Theorm, we  can  find the  radius^2  as

4^2  + 10^2  = r^2

116  = r^2

The  area of  the circle =

pi r^2  =

pi * 116  =

116  pi units^2

See the  graph here  :  https://www.desmos.com/calculator/vlrvdecxml

A = ( -10.4, -2.8)    B =  ( 5.6, 9.2)

Checkto see that AB  = 20   May 8, 2021