Mathematics
Class-X
2015-2016
Q.1. Use ruler and compass only in this question.
i. Draw a circle, with centre O and radius 4 cm.
ii. Mark a point P such that OP = 7 cm.
Construct the two tangents to the circle from P. Measure and record the length of
one of the tangents.
Solution :
A circle with centre O and radius 4 cm is drawn.
A point P is taken 7 cm from O.
O is joined with P.
OP is bisected at C.
Taking C as centre ad OC as radius a circle is drawn which intersect the previous circle at T and
T’.
PT and PT’ are joined which is the required tangents. PT = PT’ = 5.7 cm (aprox.)
Q.2. Construct an angle PQR = 45º. Mark a point S on QR such that QS = 4.5 cm.
Construct a circle to touch PQ at Q and also to pass through S.
Solution :
L PQR = 45º is drawn.
QX is drawn perpendicular to PQ at Q.
QS = 4.5 cm is taken from QR.
Perpendicular bisector of QS is drawn which intersects QX at O.
Taking O as centre and OQ as radius a circle is drawn which pass through Q and S.
Q.3. Construct a Δ ABC, in which AC = 5 cm, BC = 7 cm and AB = 6 cm.
i. Mark D, the midpoint of AB.
ii. Construct the circle which touches BC at C and passes through D.
Solution :
AB = 6 cm is drawn. From B an arc of radius 7 cm and from A an arc of radius 5 cm are drawn.
They intersect at C. Ac and BC are joined. The point D, the mid-point of AB is drawn. CD is
joined. At C perpendicular to AC is drawn. Perpendicular bisector of CD is drawn which
intersect OC at O. Taking O as centre and OD as radius, a circle is drawn. This is the required
circle to pass through C and D.
Sudheer Gupta .
Be positive and constructive. 
Page 1
Mathematics
Class-X
2015-2016
Q.4. Using ruler and compass only, construct a Δ ABC such that AB = 5 cm, LABC = 75º
and the radius of the circum-circle of triangle ABC 3.5 cm. On the same figure, construct a
circle touching AB at its middle point, and also touching the side AC.
Solution :
AB = 5 cm is drawn. Perpendicular bisector PQ of AB is drawn. Taking A as centre and radius
of 3.5 cm, an arc is drawn to intersect PQ at O. With O as centre and OA as radius, a circle is
drawn. At B an angle ABR = 75º is drawn, here BR intersect the circle at C. AC is joined. ABC
is the required triangle. Angle bisector of L BAC is drawn which intersect PQ at D. Taking D as
centre and DM as radius, another circle is drawn. It is the another required circle.
Q.5. Only ruler and compass may be used in this question,
i. Construct Δ ABC, such that AB = AC = 7 cm, BC = 5 cm.
ii. Draw AX, the perpendicular bisector of side BC.
iii. Draw a circle with centre A and radius 3 cm cutting AX at Y.
iv.
Construct another circle to touch the circle with centre A externally at Y and
passing through B and C.
Solution :
Sudheer Gupta .
Be positive and constructive. 
Page 2
Mathematics
Class-X
i.
ii.
iii.
iv.
2015-2016
BC = 5 cm is drawn. Taking B and C as centre two arcs are drawn to intersect at A. AB
and AC are joined. ABC is the required triangle.
Perpendicular bisector AD of BC is drawn.
With A as centre a circle of radius 3 cm is drawn which intersect AD at P.
PB and PC are joined. Perpendicular bisector of PC is drawn which intersect AD at O.
With O as centre and OP as radius, a circle is drawn. This is the required circle through B
and C.
Q.6. Ruler and compass only may be used in this question. All construction lines and arcs
must be clearly shown, and be of sufficient length and clarity to permit assessment.
i. Construct Δ ABC, in which AB = 9 cm, BC = 10 cm and L ABC = 45º.
ii. Draw a circle, with centre A and radius 2.5 cm. Let it meet AB at D.
iii. Construct a circle to touch the circle with centre A externally at D and also to touch
the line BC.
Solution :
AB = 9 cm is drawn. At B, L ABC = 45º is drawn. BC = 10 cm is taken. A circle with A as
centre and radius 2.5 cm is drawn which intersect AB at D. A perpendicular on AB at D is drawn
which intersect BC at E. Angle bisector of L BED is drawn which intersect DB at F. Taking F as
centre and FD as radius a circle is drawn which touch previous circle at D and the line BC at …..
Q.7. Using ruler and compass only, construct
i. A triangle ABC in which AB = 9 cm, BC = 10 cm and L ABC = 45º.
Sudheer Gupta .
Be positive and constructive. 
Page 3
Mathematics
Class-X
2015-2016
ii. construct a circle of radius 2 cm to touch the arms of L ACB in (i) above.
Solution :
AB = 9 cm is drawn. At B L ABC is drawn 45º. BC = 10 cm ia taken. AC is joined and we get Δ
ABC. Angle bisector CD of L ACB is drawn. A line parallel to BC and at a distance of 2 cm
from BC is drawn which intersect CD at D. With D as centre and radius of 2 cm a circle is drawn
which touch arms of L ACB.
Sudheer Gupta .
Be positive and constructive. 
Page 4