Question

${\Delta }_1+{\Delta }_2+{\Delta }_3$ is equal to

• A. $2\Delta$
• B. $3\Delta$
• C. $\Delta$
• D. $Noneofthese$

Answer 1

Answer: (C)

$\Delta$

$\angle ACL=\angle ABL={90}^o$ (angles in semicircle)
$\angle ACL=\angle ABC=B,\angle ALB=\angle ACB=C$
Circumcircle of triangle $BLC$ is same as that of $ABC$
${△}_1=$ Area of triangle $BLC$
$=\frac{BC\times BL\times CL}{4R}=\frac{a.c\cot C.b\cot B}{4R}$
$=\frac{{\left(2R\right)}^3.\sin A\sin B\sin C\cos B\cos C}{4R\sin B\sin C}$
$=2R^2\sin A\cos B\cos C$
Therefore, ${△}_1+{△}_2+{△}_3$
$=\frac{abc}{4R}[\cot B\cot C+\cot C\cot A+\cot A\cot B]=\frac{abc}{4R}=△$