Draw a circle with the help of a bangle Take a point outside the circle Construct the pair - Maths - Constructions

Question

Draw a circle with the help of a bangle Take a point outside the
circle Construct the pair of tangents from this point to the
circles Give the justification of the construction

Gopal Mohanty · Accepted Answer

The required tangents can be constructed on the given circle as follows Step 1 Draw a circle with the help of a bangle Step 2 Take a point P outside this circle and take two chords QR and ST Step 3 Draw perpendicular bisectors of these chords Let them intersect each other at point O Step 4 Join PO and bisect it Let U be the mid-point of PO Taking U as centre, draw a circle of radius OU, which will intersect the circle at V and W Join PV and PW PV and PW are the required tangents

Justification The construction can be justified by proving that PV and PW are the tangents to the circle For this, first of all, it has to be proved that O is the centre of the circle Let us join OV and OW

We know that perpendicular bisector of a chord passes through the centre Therefore, the perpendicular bisector of chords QR and ST pass through the centre It is clear that the intersection point of these perpendicular bisectors is the centre of the circle ∠PVO is an angle in the semi-circle We know that an angle in a semi-circle is a right angle ∴ ∠PVO = 90° ⇒ OV ⊥ PV Since OV is the radius of the circle, PV has to be a tangent of the circle Similarly, PW is a tangent of the circle

Draw a circle with the help of a bangle. Take a point outside the circle. Construct the pair of tangents from this point to the circles. Give the justification of the construction.