Question

$O$ is the centre of the circle in the figure given above. The correct relationship between $a,b$and$c$ is

• A. $a+b=c$
• B. $a=b+c$
• C. $a+c=b$
• D. $b-c-a=0$

Answer 1

The correct option is B $a=b+c$

$\angle AOB=2\angle AEB=2\angle AFB$(Central Angle is twice angle inscribed by chords)

$\therefore \angle AOB=\angle AEB+\angle AFB$

$\therefore \angle a=\angle 4+\angle 3=2\angle 4$......(1)

Now $\angle 1=\angle 2$ (Angles in the same segment)......(2)

$\angle b=\angle 4+\angle 1$ (Exterior angle =sum of two opposite interior angles)

$\Rightarrow \angle b=\angle 4+\angle 2$....(3) [from (2)]

$\angle c=\angle 4-\angle 2$...(4)[Since$(\angle c+\angle 2=\angle 4)$] Adding (3) and (4)

$\angle b+\angle c=2\angle 4=\angle a$ [From(1)]

$\angle a=\angle b+\angle c$

Hence option(B) is right option.