Question

Draw a circle of radius 3cm.Draw tangents to the circle from a point 8cm away from the centre of the circle.

Answer 1

Hint: Use the fact that the circumcircle of a right-angled triangle has a diameter as the hypotenuse of the triangle.The tangent and the radius of the circle at the point of contact are perpendicular to each other. Observe that the line joining the centre to the point from which the tangents are drawn, the radius at the point of contact and the tangent line will form a right-angled triangle. Hence the line joining the centre to the point from which tangents are drawn will be the hypotenuse of the right angled triangle. Since the circumcentre of a right angled triangle is the midpoint of its hypotenuse, we can locate the circumcentre by drawing a perpendicular bisector to the line segment joining the centre to the point from which tangents need to be drawn. Hence we can draw the circumcircle of the right-angled triangle. Since this circumcircle will meet the original circle at point of contact, we can locate the points at which the tangents touch the circle. Hence we can construct the tangents.

Complete step-by-step answer:

Steps of construction:
[1] Mark point A on the paper.
[2] Taking A as centre and radius 3cm draw a circle.
[3] Draw a line segment AC=8cm.
[4] Draw perpendicular bisector GH of AB and let it intersect AB at D.
[5] With D as centre and radius AD draw a circle. Let it intersect the first circle at E and F.
[6] Join EC and FC.
[7] EC and FC are the required tangents

Note: Justification of the construction:
We know that angle in a semicircle is right.

Hence AEC is a right-angled triangle right angled at E.
Hence AE and EC are perpendicular to each other. Since EA is the radius of the circle, we have CE is tangent to the circle.
Similarly, CF is a tangent to the circle.
Hence the constructed line segments are tangents to the circle.
[2] Observe that CE = CF, since tangents drawn from an external point to a circle are equal in length
[3] By using Pythagoras theorem we can verify tangents length CE and CF.