Question

In the figure, o is the centre of the circle seg AB is the diameter. Tangent at A intersects the secant BD at C . Tangent HD intersects side AC at J then prove seg AJ = seg CJ

Answer 1

Answer:

Step-by-step explanation:

Given : O is the centre of the circle.

Seg AB is a diameter, CD is the tangent at C.

BD is a tangent at B

In ΔCOD and ΔBOD

⇒ OC=OB    ....(Radii of same circle)

⇒CD=DB   ....(Tangent from an external point)

⇒OD=OD    ....(Common side)

⇒ ΔCOD=ΔBOD      (SSS test)

⇒ ΔCOD=ΔBOD=x     (c.a.s.t.) equation (i)

In ΔAOC,  AO=OC   ....(Radii of same circle)

∠OAC=∠OCA=y  ....(Isoceles Δ theorem) equation (ii)

⇒∠COB=∠OAC+∠OCA   ....(Remote interior ∠ theorem)

⇒∠COD+∠BOD=∠OAC+∠OCA

⇒x+x=y+y   ....[From equation (i)  and (ii)]

⇒2x=2y

⇒x=y   ....(dividing by 22)

⇒ ∠COD=∠OCA

⇒  Seg OD∥ chord AC.   ....(Alternate ∠s test)  

⇒  OD = AC      

Hence, proved.