Construct the tangent to a circle at a point outside it
Construct the tangent to a circle at a point outside it
CONSTRUCT THE TANGENT TO A CIRCLE AT A POINT OUTSIDE IT:
Case (a) : To draw the tangent to a circle at a given point on it, when the centre of the circle is known :
Required : To draw the tangent to the circle at P.
Steps of Construction:
(i) Take a point O as centre and draw a circle of given radius.
(ii) Take a point P on the circle.
(iii) Join OP.
(iv) Draw a line AB perpendicular to OP at the point P.APB is the required tangent at P.
Case (b) : To draw the tangent to a circle at a given point on it, when the centre of the circle is not known.
Given : A circle and a point P on it.
Required : To draw the tangent to the circle at P.
Steps of Construction:
(i) Draw any chord PQ through P and join P and Q to a point R in major arc PQ (or minor arc PQ).
(ii) COnstruct equal to
and on opposite side of R with respect to chord PQ.
Produce BP to A to get the required tangent APB
CONSTRUCTION OF A TANGENT TO A CIRCLE FROM A POINT OUTSIDE THE CIRCLE:
In this situation, when the point lies outside the circle, there will be two tangents to the circle from this point.
In this construction, again we have two different situations:
Case (a) : To draw the tangent to a circle from a point outside it (external point), when its centre is known.
Given : A circle with centre O and a point P outside it.
Required: To construct the tangents to the circle from point P.
Steps of Construction:
(i) Join OP and bisect it. Let M be the mid-point of OP.
(ii) Taking M as centre and MO as radius, draw a circle which intersect the given circle in two points, say A and B.
(iii) Join PA and PB. These are the required tangents from P to given circle.
Example : Draw a circle of radius 2.5 cm. From a point P, 6 cm. apart from the centre of the circle, draw two tangents to the circle.
SOLUTIONS:
Given : A point P is at a distance of 6 cm. From the centre of a circle or radius 2.5 cm.
Required : To draw two tangents to the circle from the given point P.
Steps of construction:
(i) Draw a line segment OP of length 6 cm.
(ii) With centre O and radius equal to 2.5 cm, draw a circle.
(iii) Draw the right bisector of OP, intersecting OP at M. Let M be mid-point of OP.
(iv) Taking M as centre and MO as radius draw a circle which intersect the given circle in two points, say A and B.
(v) Join PA and PB. These are the required tangents from P to the given circle.
Steps of construction :
(i) Let P be the external point from where the tangents are to be drawn to the given circle. Through P draw a secant PAB to intersect the circle at A and B (say).
(ii) Produce AP to a point C such that AP = PC i.e., P is the mid-point of AC.
(iii) Draw a semi-circle with BC as diameter.
(iv) Draw intersecting the semi-circle at D.
(v) With P as centre and PD as radius draw arcs to intersect the given circle at T and T1.
(vi) Join PT and PT1. Then PT and PT1 are the required tangents.
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