Explain the steps to construct tangents to a circle from a point P outside the circle and prove that they are tangents to the circle.
The steps for construction of a tangent from a point outside the circle are:
a) Draw a line segment from P to the centre of the circle O and bisect it. Let the bisection is done at a point, say M.
b) With PM as radius draw a circle. Let this circle intersect the initial circle at points X and Y
c) Join PX and PY. They are the required tangents.
Since PO is the diameter of the new circle, ∠ X and ∠ Y are 90∘( Angles in a semicircle)
But OX and OY are radii of the circle.
Any line that makes a right angle with a radius and touches the circle at only 1 point is a tangent to a circle.
Hence PX and PY should be tangents to the circle.