Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only construct:
(i) A circle of radius 2.5 cm passing through A and C.
(ii) Construct two tangents to the circle from the external point B.
Measure and record the lengths of the tangents.
Answer
Steps of construction :
Draw a line segment AB = 5 cm.
With A as center cut an arc of 3 cm on AB to obtain C.
With A as center and radius = 2.5 cm draw an arc.
With C as center and radius = 2.5 cm draw an arc cutting the previous arc and mark the point O.
With O as center and radius = 2.5 cm draw a circle.
Join OB.
Draw the perpendicular bisector of OB, let it meet OB at point M.
With M as center and radius equal to OM, draw a circle to cut the previous circle at points P and Q.
Join PB and QB. Hence, PB and QB are required tangents.
Measure PB and QB.
On measuring, PB = QB = 3.2 cm.
Hence, length of each tangent = 3.2 cm.
Draw a circle of radius 3 cm. Take a point P on it. Using ruler and compasses only construct a tangent to the circle at the point P.
Answer
Steps of Construction :
Draw a circle of radius = 3 cm with centre as O.
Extend the radius of the circle on one the side and mark point P, Join OP.
At point P, construct a line perpendicular to OP and name it as TT'.
Draw a circle of radius 3.4 cm. Take a point P on it. Without using the centre of the circle, construct a tangent to the circle at the point P.
Answer
Steps of Construction :
Draw a circle of radius = 3.4 cm with centre as O.
Extend the radius of the circle on one the side and mark point P, Join OP.
At point P, construct a line perpendicular to OP and name it as L.
Draw a circle of radius 2.7 cm. Mark its centre as O. Take a point P at a distance of 5.3 cm from O. From the point P, draw two tangents to the circle. Measure the length of each.
Answer
Steps of Construction :
Draw a circle of radius = 2.7 cm with centre as O.
Mark a point P at a distance of 5.3 cm from O. Join OP. Draw its perpendicular bisector to meet OP at M.
With M as centre and OM as radius, draw a circle. Let this circle intersect the given circle with centre as O at points A and B.
Join PA and PB.
On measuring we get : PA = PB = 4.7 cm.
Hence, the length of each tangent = 4.7 cm.
Draw a circle of radius 4 cm. Mark its centre as C and mark a point D such that CD = 7 cm. Using ruler and compasses only but not using the centre of the circle, construct two tangents from D.
Answer
Steps of construction :
Construct a circle with centre as C and radius = 4 cm.
Mark a point D at a distance of 7 cm from the centre. Join CD. Draw its perpendicular bisector to meet CD at M.
With M as centre and CM (or MD) as radius, draw a circle. Let this circle intersect the given circle with centre as C at points A and B.
Join DA and DB.
Draw a circle of radius 3.2 cm. Draw two tangents to it inclined at an angle of 60° with each other.
Answer
Steps of Construction :
Draw a circle with center O, radius = 3.2 cm and BC as diameter.
Draw arcs making an angle of 120° at O such that ∠AOB = 120°.
At A and B, draw two rays making an angle of 90° at each point which meet other at point P.
Hence, AP and BP are required tangents making an angle of 60° with each other.
Draw a circle of radius 2.5 cm. Draw two tangents to it inclined at an angle of 45° to each other.
Answer
Steps of Construction :
Draw a circle with center O, radius = 2.5 cm and BC as diameter.
Draw arcs making an angle of 135° (180° - 45°) at O such that ∠AOB = 135°.
At A and B, draw two rays making an angle of 90° at each point which meet other at point P.
Hence, AP and BP are required tangents making an angle of 45° with each other.
Draw a circle of radius 3.5 cm. Mark a point P outside the circle at a distance of 6 cm from the centre. Construct two tangents from P to the given circle. Measure the length of one tangent.
Answer
Steps of construction :
With O as center and radius = 3.5 cm draw a circle.
Take a point P such that OP = 6 cm.
Draw perpendicular bisector of OP, cutting OP at M.
With center M and radius OM, draw a circle which intersects the circle with center O at A and B.
Join AP and BP. Measure AP and BP.
Hence, AP and BP are required tangents.
On measuring,
AP = BP = 4.9 cm.
Hence, length of each tangent = 4.9 cm.
Use ruler and compasses 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 :
Construct a circle with centre as O and radius = 4 cm.
Mark a point P at a distance of 7 cm from the centre. Join OP. Draw its perpendicular bisector to meet OP at M.
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.
Join PA and PB.
On measuring, PA = PB = 5.7 cm
Hence, the length of each tangent = 5.7 cm.