Ex.10.2 Q4 Circles Solution - NCERT Maths Class 10

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Question

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Text Solution

What is known?

  • Let \(AB\) be a diameter of the circle.
  • Two tangents \(PQ\) and \(RS\) are drawn at points \(A\) and \(B\) respectively.

To Prove:

Tangents drawn at the ends of a diameter of a circle are parallel.

Reasoning :

  • A tangent to a circle is a line that intersects the circle at only one point.
  • Theorem 10.1 : The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Steps:

We know that, according to Theorem 10.1, Radius is perpendicular to the tangent at the point of contact

Thus, \({OA}\, \bot \, {PQ} \) and \({OB}\, \bot\, {Z}\)

Since the Tangents are Perpendicular to the Radius,

\[\begin{aligned} \angle {P A O} & = 90 ^ { \circ } \\ \angle {R B O} & = 90 ^ { \circ } \\ \angle {O A Q} & = 90 ^ { \circ } \\ \angle {O B S} & = 90 ^ { \circ } \end{aligned}\]

Here \(\angle {OAQ}\) & \(\angle {OBR}\) and \(\angle {PAO}\) & \(\angle {OBS}\) are two pairs of alternate interior angles and they are equal.

If the Alternate interior angles are equal then lines \(PQ\) and \(RS\) should be parallel.

We know that \(PQ\) & \(RS\) are the tangents drawn to the circle at the ends of the diameter \(A B.\)

Hence, it is proved that Tangents drawn at the ends of a diameter of a circle are parallel.