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Question Number 76112 by Maclaurin Stickker last updated on 23/Dec/19
If the sides of a triangle are consecutive integers and the maximum angle is twice the minimum, determine the sides of the triangle.
Ifthesidesofatriangleareconsecutiveintegersandthemaximumangleistwicetheminimum,determinethesidesofthetriangle.
Answered by mr W last updated on 24/Dec/19
say the sides are n−1,n,n+1 cos θ_(max) =(((n−1)^2 +n^2 −(n+1)^2 )/(2(n−1)n))=((n−4)/(2(n−1))) cos θ_(min) =(((n+1)^2 +n^2 −(n−1)^2 )/(2(n+1)n))=((n+4)/(2(n+1))) θ_(max) =2θ_(min) cos θ_(max) =2 cos^2 θ_(min) −1 ((n−4)/(2(n−1)))=2×(((n+4)^2 )/(4(n+1)^2 ))−1 ((n−4)/(n−1))=((4n+14−n^2 )/(n^2 +2n+1)) 2n^3 −7n^2 −17n+10=0 (n+2)(n−5)(2n−1)=0 ⇒n=5 ⇒sides are 4, 5 and 6.
saythesidesaren−1,n,n+1cosθmax=(n−1)2+n2−(n+1)22(n−1)n=n−42(n−1)cosθmin=(n+1)2+n2−(n−1)22(n+1)n=n+42(n+1)θmax=2θmincosθmax=2cos2θmin−1n−42(n−1)=2×(n+4)24(n+1)2−1n−4n−1=4n+14−n2n2+2n+12n3−7n2−17n+10=0(n+2)(n−5)(2n−1)=0⇒n=5⇒sidesare4,5and6.
Commented by Maclaurin Stickker last updated on 24/Dec/19
I did by the same way, but I wasn′t sure. Thank you for the great answer.
Ididbythesameway,butIwasn′tsure.Thankyouforthegreatanswer.

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