>
Prove that the line segment joining the point of contact of two parallel tangents to a circle is a diameter of the circle.
Solution
Given : CD and EF are two parallel tangents at the points A and B of a circle with center O.
To prove : AOB is a diameter of the circle
Construction : Join OA and OB
Draw OG | | CD
Proof : OG | | CD and AO cuts them .
⇒ 90∘ + GOA = 180∘ [ OA is perpendicular to CD ]
⇒ GOA = 90∘)
Similarly, GOB = 90∘;
Therefore, GOA + GOB = (90∘ + 90∘) = 180∘)
=> AOB is a straight line
Hence, AOB is a diameter of the circle with center O.
11
>
Prove that the line segment joining the point of contact of two parallel tangles of a circle passes through its centre.
Solution
Let XBY and PCQ be two parallel tangents to a circle with centre O.
Construction: Join OB and OC.
Draw OA || XY
⇒∠XBO+∠AOB=180∘ (sum of adjacent interior angles is 180∘)
Now, ∠XBO=90∘ (A tangent to a circle is perpendicular to the radius through the point of contact)
⇒90∘+∠AOB=180∘
⇒∠AOB=180∘−90∘=90∘
Similarly, ∠AOC=90∘
∠AOB+∠AOC=90∘+90∘=180∘
Hence, BOC is a straight line passing through O.
Thus, the line segment joining the points of contact of two parallel tangents of a circle passes through its centre.
NCERT Previous Years Papers
Suggest Corrections
16