The sides of a triangle are three consecutive integers and its inradius is 4. Find the circumradius.
Inradius = Area / Semiperimeter
Sides are: x-1, x, x+1
Semiperimeter = 3x/2
Area = sqrt( 3x/2 * (x/2+1) * x/2 * (x/2-1))
Area = sqrt( 3x^2/4 * (x^2/4 - 1) )
Area = sqrt( 3x^4/16 - 3x^2/4 )
Area = sqrt( 3x^4/16 - 12x^2/16 )
4 = sqrt( 3x^4/16 - 12x^2/16 ) / (3x/2)
4 = 2 (x/4) * √3 * sqrt( x^2 - 4 ) / 3x
8 √3 = sqrt(x^2 - 4)
192 = x^2 - 4
x = 14
Sides: 13, 14, 15
Circumradius of triangle whose sides are a,b,c and area is A: abc/4A
Area = sqrt( 21 * 8 * 7 * 6) = √7056 = 84
Circumradius = 13*14*15/(4*84) = 65/8 = 8.125
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Posted by Larry
on 2024-05-15 12:12:32 |
Analytic solution
