Geometry Construction

Exercise 11.2

Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

Solution: Draw a circle with radius 6 cm and centre O.

Draw a line segment OP = 10 cm
Make perpendicular bisector of OP which intersects OP at point O’.
Take O’P as radius and draw another circle.
From point P, draw tangents to points of intersection between the two circles.
PQ = PR = 8 cm.

Justification: Radius, tangent and distance between centre and external point (from which tangent is drawn) make a right triangle. Using Pythagoras theorem, we have;

`PQ^2 = OP^2 – OQ^2`

`= 10^2 – 6^2`

`= 100 – 36 = 64`

Or, `PQ = 8 cm`

Question 2: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

Solution: Draw two concentric circles with radii 4 cm and 6 cm.

Draw the radius OP of bigger circle.
Make a perpendicular bisector of OP which intersects OP at point O’.
Taking O’P as radius, draw another circle.
From point P, draw tangents PQ and PR; as shown in figure.

Justification: Using Pythagoras theorem, we have;

`PQ^2 = OP^2 – OQ^2`

`= 6^2 – 4^2`

`= 36 – 16 = 20`

Or, `PQ = 2sqrt5 cm`

Question 3: Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

Solution: Draw a circle with radius 3 cm.

Extend the diameter to A and B on either side.
Draw perpendicular bisectors of OA and OB.
Perpendicular bisector of OA intersects it at point O’.
Perpendicular bisector of OB intersects it at point O”.
Taking O’A as radius, draw the second circle.
Taking O”B as radius, draw the third circle.
From point A, draw tangents AP and AQ.
From point B, draw tangents BR and BS.

Question 4: Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60^o.

Solution: Draw a circle with radius 5 cm.

Draw a line AP perpendicular to radius OA.
Draw OB at 120^o from OA. (Because angle at centre is double the angle between two tangents).
Join A and B to P; to get two tangents.
Here, ∠APB = 60^o

Question 5: Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Solution: Draw a line segment, AB = 8 cm.

Taking A as centre, draw a circle with radius 4 cm.
Taking B as centre, draw another circle with radius 3 cm.
Draw perpendicular bisector of AB.
Taking midpoint of AB as centre and AB as diameter, draw the third circle.
From point A, draw tangents AR and AS.
From point B, draw tangents BP and BQ.

Question 6: Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and ∠B = 90^o. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Construct the tangent from A to this circle.

Solution: Draw a line segment AB = 6 cm.

Make a right angle at point B and draw BC = 8 cm.
Draw a perpendicular BD to AC.
We know that in a right triangle, hypotenuse is the diameter of circumcircle.
Taking BC as diameter, draw a circle which passes through points B, C and D.
Join A to O and taking AO as diameter, draw second circle.
From point A, draw tangents AB and AP.

Question 7: 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 circle.

Solution: This question can be solved as the first question.