### Mathematics Class X

Real Numbers
Polynomials
Arithmetic Progressions
Triangles
Coordinate Geometry
Introduction to Trigonometry
Circles
Constructions
Areas Related to Circles
Surface Areas and Volumes
Statistics
Probability

# Tangent to a Circle

All the tangents of a circle are perpendicular to the radius through the point of contact of that tangent.

OP is the radius of the circle and Q is any point on the line XY which is the tangent to the circle. As OP is the shortest line of all the distances of the point O to the points on XY. So OP is perpendicular to XY. Hence, OP XY

Example

Find the radius of the circle, if the length of the tangent from point A which is 5 cm away from center is 4 cm.

Solution

As we know that the radius is perpendicular to the radius, so the ABO is a right angle triangle.

Given, AO = 5 cm and AB = 4 cm We can use Pythagoras theorem here

OA2 = OB2 + AB2

OB= OA2 – AB2

= 52 – 42

= 25 – 16

OB= 9

OB = 3 So the radius of the given circle is 3 cm

Example

If two tangents PA and PB are drawn to a circle from a point P with centre O and OP is equal to the diameter of the circle then show that triangle APB is an equilateral triangle.

Solution

Given, is a tangent to the circle.

Therefore, (Tangent is perpendicular to the radius through the point of contact) In , So Also Now, In (length of the tangents from the external point is equal) (Angles opposite to equal sides are equal) (Due to angle sum property)   As Hence, is an equilateral triangle.

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