Circumscribed 83363

Triangle ABC, with sides a = 15 cm, b = 17.4 cm, and c = 21.6 cm, is circumscribed by a circle. Calculate the area of the segments determined by the sides of the triangle.

Correct answer:

S1 =  30.9391 cm2
S2 =  52.8996 cm2
S3 =  158.1036 cm2

Step-by-step explanation:

a=15 cm b=17.4 cm c=21.6 cm  a2 = b2+c2  2bc   cos α  A=arccos(2 b cb2+c2a2)=arccos(2 17.4 21.617.42+21.62152)0.761 rad α=A  °=A π180   °=0.761 π180   °=43.60282  ° B=arccos(2 a ca2+c2b2)=arccos(2 15 21.6152+21.6217.42)0.9273 rad β=B  °=B π180   °=0.9273 π180   °=53.1301  ° γ=180αβ=18043.602853.130183.2671   r=2 sinαa=2 sin43.602818972704° a=2 sin43.602818972704° 15=2 0.68965515=10.875 cm  x1=r2(a/2)2=10.8752(15/2)2=863=7.875 cm x2=r2(b/2)2=10.8752(17.4/2)2=40261=6.525 cm x3=r2(c/2)2=10.8752(21.6/2)2=4051=1.275 cm  h1=rx1=10.8757.875=3 cm h2=rx2=10.8756.525=2087=4.35 cm h3=rx3=10.8751.275=548=9.6 cm  a2 = r2+r2  2r2   cos φ1 φ1=arccos(2 r22 r2a2)=arccos(2 10.87522 10.8752152)1.522 rad φ2=arccos(2 r22 r2b2)=arccos(2 10.87522 10.875217.42)1.8546 rad φ3=arccos(2 r22 r2c2)=arccos(2 10.87522 10.875221.62)2.9066 rad  S0=π r2=3.1416 10.8752371.5424 cm2  S1=S0 2πφ121 a x1=371.5424 2 3.14161.52221 15 7.875=30.9391 cm2
S2=S0 2πφ221 b x2=371.5424 2 3.14161.854621 17.4 6.525=52.8996 cm2
S3=S0 2πφ321 c x3=371.5424 2 3.14162.906621 21.6 1.275158.1036 cm2   Verifying Solution:  t1=φ1+φ2+φ3=1.522+1.8546+2.90666.2832 rad s=2a+b+c=215+17.4+21.6=27 cm S=s (sa) (sb) (sc)=27 (2715) (2717.4) (2721.6)=5648=129.6 cm2 S4=S1+S2+S3+S=30.9391+52.8996+158.1036+129.6371.5424 cm2 S4=S0

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