Draw a circle of radius 4cm.Take a point P outside the circle.Without using the centre of the circle,draw 2 tangents to the circle from point P.Please write the justification for the above construction.
Step 1 Draw a circle of radius 4 cm
Step 2 Take a point P outside this circle and take two chords AB and CD.
Step 3 Draw perpendicular bisectors of these chords. Let them intersect each other at point O.
Step 4 Join PO and bisect it. Let M be the mid-point of PO. Taking M as centre, draw a circle of radius OM, which will intersect the circle at T and T'. Join PT and PT'.
PT and PT' are the required tangents.Justification - firstly, O is the center of the circle, because we know that the perpendicular bisector of a chord passes through the center and hence, the perpendicular bisectors of 2 chords will always intersect at the center only. Furthermore, to prove that PT and PT' will be tangents, we know that the angle in a semi-circle is 90 degree. Hence, angle (PTO) will be 90 degree (considering the circle with M as the center and OM as the radius) and hence, PT and PT' will be tangents.