Question

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.​

Answer 1

the figure is also linked

Step-by-step explanation:

given :- O is cenet of circle

to prove :- angleABC = 1/2 angleAOC

contruction :- Diameter of circle strating from B and going through O to meet D

proof:-

In triangle AOB

AO=OB ...........(radii of same circle)

angleOAB = angleOBA ...(angle opposite to equal sides)

consider angleOAB and angleOBA as x ..........1

In triangleBOC

BO=OC ............(radii of same circle)

angleOCB = angleOBC .......( angle oppsite to equal sides)

consider angleOCB and angleOBC as y ..............2

As angleAOD is an exterior angle of triangleAOB

x+x =vangleAOD ..........(by 1 and exterior angle theorem)

angleAOD = 2x ..............3

As angleDOC is an exterior angle of triangleBOC

y+y = angleDOC

angleDOC = 2y .........(by 2 and exterior angle theorem) ............4.

angleABC = angleABO + angleOBC

angleABC = x+y .......(by 1 and2) .........5

angleAOC = angleAOD + angleDOC

angleAOC = 2x + 2y

angleAOC = 2(x+y)

i.e. angleAOC = 2(angleABC)

thus angleABC=1/2 angleAOC