Question

In adjoining figure o is the center of the circle.seg AB is diameter.at point CD is drawn from line BD is a tangent to the circle at point B then show that seg OD llchord AC​

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 2)

⇒∠COD=∠OCA

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

Hence, proved.