Heron backlaw

Calculate the missing side in a triangle with sides 25 and 13 and area 152.

Correct answer:

x1 =  23.7516
x2 =  31.9978

Step-by-step explanation:

a=25 b=13 S=152 s=  (a+b+c)/2 S= s(sa)(sb)(sc)  S2 =  (a+b+c)/2 ( (a+b+c)/2a) ( (a+b+c)/2b) ( (a+b+c)/2c) 16 S2 =  (a+b+c) ( (a+b+c)2a) ( (a+b+c)2b) ( (a+b+c)2c) 16 S2 =  (a+b+c) (b+ca) (ab+c) (a+bc)  16 S2 = a4 + 2 a2 b2  b4 + 2 a2 c2 + 2 b2 c2  c4  S = 2a h = 2a b sin α  α=arcsin(a b2 S)=arcsin(25 132 152)1.2093 rad  x1=a2+b22 a b cosα=252+1322 25 13 cos1.209323.7516   Verifying Solution:  s1=2a+b+x1=225+13+23.751630.8758 S1=s1 (s1a) (s1b) (s1x1)=30.8758 (30.875825) (30.875813) (30.875823.7516)=152 S1=S
sin α= sin(π  α)  α2=πα=3.14161.20931.9322 rad  x2=a2+b22 a b cos(α2)=252+1322 25 13 cos1.932231.9978 s2=2a+b+x2=225+13+31.997834.9989 S2=s2 (s2a) (s2b) (s2x2)=34.9989 (34.998925) (34.998913) (34.998931.9978)=152 S2=S

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