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# help

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A convex quadrilateral is inscribed in a circle so that one of its sides is a diameter of the cirlce.  The length of the two sides of the quadrialteral which have exactly one endpoint on this diameter are 7 and 20.  If the length of the longest side of this quadrilateral is 25, find the length of the fourth side of the quadrilateral.

Dec 10, 2019

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See  the following image   : We have a  circle with the equation x^2 + y^2  = 12.5^2

And

AB  = 25    BC  = 7

Then triangle  ABC  is a 7 - 24 - 25  right triangle

The area of this triangle  =   7 * 24  / 2    =  84 units^2

And we can find the height of this triangle as

84  =  (1/2)(25) h

84 / 12.5  = h  =   168/25  =   6.72  units  =  the y value  of  point C

So....the x value of point C is given as     x = -  sqrt  [12.5^2 - 6.72^2 ]  =  -10.54

Then triangle ADB will also  be a right triangle  with sides   15 - 20  -  25

And the area of this triangle  =  (1/2)(15)(20)  = 150

And we can find the height of this triangle as

150  = (1/2) 25  * h

150/12.5  =  h   =   12  =  the y coordinate of  point  D

So  the  x coordinate  of D  is given as    x  =  - sqrt [ 12.5^2 - 12^2]  = -3.5

So.....the length  of the missing side  =  chord CD   =

sqrt  [ (-10.54  + 3.5)^2  +  ( 12 - 6.72)^2 ]    =  8.8 units   Dec 11, 2019