Question #146619
Line l touches the circle ω at point K. Points A and B are chosen on ω in such a way that they are situated on opposite sides with respect to the diameter of ω that passes through point K. Find the area of triangle AKB if the distances from points A and B to line l are equal to 7 and 11 respectively, and AK=13. Round the answer to the closest integer.
1
Expert's answer
2020-11-26T19:11:30-0500

Let Fl,AFl;Tl,BTl;NBT,ANBT;F\in l, AF\perp l; T\in l, BT\perp l; N\in BT, AN\perp BT; O is the center of the circle ω.

AF=7,AK=13,BT=11.AF=7, AK=13, BT=11.

AKF:KF=AK2AF2=13272=12010.95.\triangle AKF: KF=\sqrt{AK^2-AF^2}=\sqrt{13^2-7^2}=\sqrt{120}\approx 10.95.

AKF=arcsinAFAK=arcsin713.\angle AKF=arcsin\frac{AF}{AK}=arcsin\frac{7}{13}.

AOK:AOK=2AKF=2arcsin713.\triangle AOK: \angle AOK=2\angle AKF=2arcsin\frac{7}{13}.

OB=OA=OK=AK2sinAOK2=132713=16914.OB=OA=OK=\frac{AK}{2sin\frac{\angle AOK}{2}}=\frac{13}{2\cdot \frac{7}{13}}=\frac{169}{14}.

KBT:BKT=arcsinBTBK=arcsin11BK.\triangle KBT: \angle BKT=arcsin\frac{BT}{BK}=arcsin\frac{11}{BK}.

OBK:BOK=2BKT=2arcsin11BK.\triangle OBK: \angle BOK=2\angle BKT=2arcsin\frac{11}{BK}.

BK=2OKsinBOK2=21691411BK,BK=2OKsin\frac{\angle BOK}{2}=2\cdot \frac{ 169}{14}\cdot \frac{ 11}{BK},

BK=16911716.30.BK=\sqrt{\frac{169\cdot11}{7}}\approx16.30.

KBT:TK=BK2BT2=16.3211212.03.\triangle KBT: TK=\sqrt{BK^2-BT^2}=\sqrt{16.3^2-11^2}\approx 12.03.

AN=TF=TK+KF=12.03+10.95=22.98.AN=TF=TK+KF=12.03+10.95=22.98.

AF=NT=7,BN=BTNT=117=4.AF=NT=7, BN=BT-NT=11-7=4.

ANB:AB=AN2+BN2=22.982+4223.33.\triangle ANB: AB=\sqrt{AN^2+BN^2}=\sqrt{22.98^2+4^2}\approx 23.33.

And by the Heron's formula, S=102.35102.S=102.35\approx 102.


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