Constructions — Exercise 16.1

Use ruler and compass only for answering this question.

Draw a circle of radius 4 cm. Mark the centre as O. Mark a point P outside the circle at a distance of 7 cm from the centre. Construct two tangents to the circle from the external point P. Measure and write down the length of any one tangent.

Answer

Steps of construction :

1. Construct a circle with centre as O and radius = 4 cm.

2. Mark a point P at a distance of 7 cm from the centre. Join OP. Draw its perpendicular bisector to meet OP at M.

3. With M as centre and OM (or MP) as radius, draw a circle. Let this circle intersect the given circle with centre as O at points A and B.

4. Join PA and PB.

Thus, PA and PB are tangents to the circle with centre O. On measuring, the length of tangent = 5.7 cm.

Draw a line AB = 6 cm. Construct a circle with AB as diameter. Mark a point P at a distance of 5 cm from the mid-point of AB. Construct two tangents from P to the circle with AB as a diameter. Measure the length of each tangent.

Answer

1. Draw a line segment AB = 6 cm.

2. Mark the mid-point of AB as O. Now as radius OA = 3 cm or OB = 3 cm construct a circle.

3. Mark a point P at a distance of 5 cm from O. Join OP. Draw its perpendicular bisector to meet OP at M.

4. With M as centre and OM (or MP) as radius, draw a circle. Let this circle intersect the given circle at points C and D.

5. Join PC and PD.

Thus, PC and PD are tangents to the circle with diameter AB. On measuring, the length of tangent = 4 cm.

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.

Answer

Steps of Construction :

1. Draw two concentric circles of radii 4 cm and 6 cm with point O as their centre.

2. Let P be a point on the outer circle. Join OP and draw its perpendicular bisector to meet OP at M.

3. Taking M as centre and OM (or MP) as radius, draw a circle. Let the circle intersect the smaller circle i.e. circle of radius 4 cm at points A and B.

4. Join PA and PB. Then PA and PB are the required tangents. On measuring, PA (or PB), we find that PA = 4.5 cm.

Calculation of length of PA

Join OA

In △OAP, ∠OAP = 90° (As angle in semicircle = 90°.)

By pythagoras theorem we get,

     OA² + PA² = OP²
⇒ PA² = OP² - OA²
⇒ PA² = 6² - 4² = 36 - 16 = 20 cm.

PA = $\sqrt{20}$ = 4.5 cm.

Hence, the length of tangent = 4.5 cm.

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.

Answer

Steps of Construction :

1. Draw a circle of radius = 3 cm with centre as O.

2. Extend the diameter (AB) of the circle on both the sides and mark points P and Q on opposite sides such that OP = OQ = 7 cm.

3. Draw perpendicular bisector of OP and OQ such that they meet OP and OQ at M and N respectively.

4. Construct circle with M as centre and OM or MP as radius. Let this circle touch the circle with centre O at points C and D.

5. Construct circle with N as centre and ON or NQ as radius. Let this circle touch the circle with centre O at points E and F.

6. Join PC, PD, QE and QF.

Hence, PC, PD, QE and QF are the tangents from point P and Q to the circle with centre as O.