Question

It has been asked that you have to draw a circle of radius 3 cm and then you have to take two points P and Q on one of its extensions on both sides of its centre, each at a distance of 7 cm on opposite sides of its centre. After that you have to draw tangents to the circle from each of these two points P and Q.

Answer 1

Hint: In this question first we have to draw a circle of the given radius and then we have to draw a secant that will pass through the centre of the circle and have to mark two points on it such that points are at equal distance from the centre of the circle. After this we will just find the midpoint of the distance between one of the points and centre of the circle using the perpendicular bisector of the distance. Then by using mid-point as centre and radius equal to distance between midpoint and centre of the circle we have to draw two arcs which will intersect the main circle at two points. Then we will join these new points with the previous point on the secant to get two tangents to the circle and we will repeat the above procedure for the other point on the opposite side of the centre.

Complete step-by-step answer:
To construct/draw the required, we have to follow certain steps which we will call as the steps of construction:
Step 1: First, draw a circle using O as centre and the radius $ = 3cm$ using the compass.

Step 2: Now draw a secant that will pass through the centre O.

Step 3: After that, mark two points P and Q on the opposite sides of the centre O each at a distance of 7cm from the centre O.

Step 4: Now, place the compass on point P and draw two arcs on the opposite sides of OP. Now place the compass on O and draw another two arcs which will intersect the previous arcs drawn from point P.

Step 5: Now, join the intersection points of the arcs to get the perpendicular bisector of OP. Mark the point where the perpendicular bisector bisects the OP as M1 also known as the midpoint of OP.

Step 6: From M1 draw another circle with radius = M1P = M1O such that this circle intersects the main circle at points A and B.

Step 7: Join P – A and P – B, such that PA and PB become the tangents to the circle at points A and B respectively.

Step 8: Now, place the compass on point Q and draw two arcs on the opposite sides of OQ. Now place the compass on O and draw another two arcs which will intersect the previous arcs drawn from point Q.

Step 9: Now, join the intersection points of the arcs to get the perpendicular bisector of OQ. Mark the point where the perpendicular bisector bisects the OQ as M2 also known as the midpoint of OQ.

Step 10: From M2 draw another circle with radius = M2Q = M2O such that this circle intersects the main circle at points C and D.

Step 11: Join Q – C and Q – D, such that QC and QD become the tangents to the circle at points C and D respectively.

Hence, PA, PB, QC and QD are the required tangents to the circle with the radius = 3cm.

Note: In such questions, we should be knowing what actually tangent is as a tangent to a circle is a straight line which touches the circle at only one point. This point is called the point of tangency. The tangent to a circle is perpendicular to the radius at the point of tangency. For such questions, we just have to follow the steps of construction and should write the steps of construction side by side with the construction so that we will get an idea what to do next. The diagram should be made with the help of a compass. We should also be very careful while working with the compass because it’s pointy and should be handled with great care.