Question #191650

Given a triangle whose sides are 42 cm, 50 cm, and 63 cm. Find the radius of a circle which is tangent to the 50 cm and 63 cm side of the triangle, and whose center lies on the third side.


1
Expert's answer
2021-05-17T15:22:08-0400

SABC=p(pa)(pb)(pc)S_{ABC}=\sqrt{\smash[b]{p(p-a)(p-b)(p-c)}}

p=a+b+c2p={a+b+c \over 2}

AB=a=50,BC=b=63,AC=c=42AB=a=50, BC=b=63, AC=c=42

p=50+63+422=1552=77.5p={50+63+42 \over 2}={155 \over 2}=77.5


SABC=77.5(77.550)(77.563)(77.542)S_{ABC}=\sqrt{\smash[b]{77.5 \ast (77.5-50) \ast (77.5-63) \ast (77.5-42)}}

SABC=1.25702119S_{ABC}=1.25 \ast \sqrt{702119}

SABC=SABO+SOBC=RAB2+RBC2S_{ABC}=S_{ABO}+S_{OBC}={R \ast AB \over 2}+{R \ast BC \over 2}


SABC=50R2+63R2=113R2S_{ABC}={50 \ast R \over 2}+{63 \ast R \over 2}={113 \ast R \over 2}


113R2=1.25702119{113 \ast R \over 2}=1.25 \ast \sqrt{702119}

R=1070211945218.5R={10 \ast \sqrt{702119} \over 452} \approx 18.5


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