Q4 of 26 Page 163

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


Let PQ be a diameter of a circle with center O, AB and CD are two tangents drawn at the ends P and Q respectively.


To Show: AB || CD


Now, we know


Tangent at any point is perpendicular to the radius at the point of contact


⇒ OP ⊥ AB and OQ ⊥ CD


⇒ ∠OPA = ∠OQD [Both 90°]


⇒ AB || CD [If two lines are cut by a transversal and the alternate interior angles are equal, then the lines are parallel]


Hence, Proved.


More from this chapter

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2

In the picture, all sides of a rhombus are tangents to a circle.


Draw this picture in your notebook.

 3

What sort of a quadrilateral is formed by the tangents at the ends of two diameters of a circle?

 1

Draw a circle of radius 2.5 centimetres. Draw a triangle of angles 40°, 60°, 80° with all its sides touching the circle.

 2

In the picture, the small (blue) triangle is equilateral. The sides of the large (red) triangle are tangents to the circumcircle of the small triangle at its vertices.


i) Prove that the large triangle is also equilateral and its sides are double that of the small triangle.


ii) Draw this picture, with sides of the smaller triangle 3 centimetres.


iii) Instead of an equilateral triangle, if we draw the tangents to the circumcircle of any other triangle at its vertices, do we get a similar triangle with double the sides? Justify.