Q25 of 49 Page 1

In Fig. 5, l and m are two parallel tangents to a circle with centre O, touching the circle at A and B respectively. Another tangent at C intersects the line l at D and m at E. Prove that ∠DOE = 90°.

jvjv.png


fig.5

nchc.png

Proof:


Let l be XY and m Be X’Y’.


From fig-


∠XDE + ∠X'ED = 180°


[∵ Sum of consecutive interior angles is 180°.]


(1/2) ∠XDE + (1/2) ∠X'ED = 90°


⇒ ∠1 + ∠2 = 90° ...(1)


[∵ OD is equally inclined at the tangents]


In Δ ODE,


(∠1 + ∠2) + ∠3 = 180°


[∵ Angle-sum-property of a Δ]


⇒ 90° + ∠3 = 180° [from (1)]


⇒ ∠3 = 90°


∴ ∠ DOE = 90° …proved


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