Mathematics [4 marks sum]

Steps of construction:
(i) Take a point O in the plane paper and draw a circle of radius 3 cm.
(ii) Mark a point P at distance 5.5 cm from the centre O and join OP.
(iii) Draw the right bisector at OP, intersecting OP at Q.
(iv) Taking Q as the centre and OQ = PQ as radius, draw a circle to intersect the given circle at T and T'.
(v) Join PT and PT' to get the required tangent.
Taype (II). Construction of a tangent to a circle from an external point when its centre is known.
Steps of construction:
Let P be the external point from where the tangent are to be drawn to the given circle.
(i) Through P draw a secant PAB to intersect the circle at A and B.

(ii) Join AP to a point C such that AP = DX is equal to the mid-point at AC.
(iii) Draw a semicircle with BC as diameter.
(iv) Draw PD ⊥ BCX intersecting the semicircle at D.
(v) With P as centre and PD as radius draw arcs to intersect the given circle at T and T'.
(vi) Join PT and PT'. Then PT and PT' are the required tangent.

Steps of construction:
1) Draw a line segment BC = 3 cm and make ∠ CBP = 45°
2) Construct EB ⊥ BP.
3) Draw the perpendicular bisector of BC its intersecting BE in O and BC in D.
4) Draw a circle taking O as a centre and OB as the radius.
5) Now with D as centre and radius = 3.5 cm. draw arcs of the circle intersecting the above-drawn circle in A and A'.
6) Join AB, AC and A'B, A'C.
Then the Δs, ΔABC and ΔA'BC are the required triangles.

Steps of construction:
1) With centre O and radius = 3 cm, draw a circle.
2) Take any point P outside the circle.
3) Through the external point, P draw a straight line PBA meeting the circle at A and B.
4) Draw a semicircle on AP as diameter.
5) Draw BC ⊥ AP, which intersects the semicircle at C.
6) With centre P and radius, PC draw an arc cutting the circle at Q.
7) Join PQ. Then PQ is the required tangent.

Steps of construction:
(i)
1) Draw a line segment BC = 6 cm.
2) Draw ∠ ABC = 120°.
3) With centre B and radius BA = 3.5 cm, draw an arc.
4) Join AB and AC.
(ii)
1) Draw the right bisector on line BC cut the BC line at M.
2) Draw another bisector of line AC.
3) With M as centre and radius MB draw a circle cut AC bisector at P.
(iii) ∠ BCP = 30°

Steps of construction:
1) Take a point O at a distance of 3.8 cm from AB and with O as centre draw a circle of radius 1.3 cm.
2) With O as centre and radius equal to 4.7 cm, draw an arc cut BA at P. Draw PX perpendicular to AB.
3) Produce AO to cut the circle at C and join CP cutting the circle at D.
4) Join OD and produce it to cut PX at S. With S as centre and radius = SD, draw the circle PDR. This is the required circle.

Steps of construction:
1) Take BC = 10 cm
2) Make ∠ ABC = 45° and with centre B, cut the arc = 9 cm.
3) Join AC, So Δ ABC is the required triangle.
4) With A as centre and radius = 2.5 cm, draw a circle. It will pass through D.
5) Draw DE ⊥ AB, which cuts BC at E.
6) Draw the angle bisector of ∠ BED which cut BD at O.
7) Taking Radius = OD to draw a circle which touches the first circle at D and also touches the line BC at F.
8) This is the required circle. The radius OD = 2.7 cm.

Steps of construction:
(i) Draw a circle of radius 4 cm.
(ii) Take a point P outside the circle and draw a secant PAB, intersecting the circle at A and B.
(iii) Produce AP to C such that AP = CP.
(iv) Draw a semicircle with CB as diameter.
(v) Draw PD and CB intersecting the semi-circle at D.
(vi) With P as centre and PD as a radius draw arcs to intersect the given circle at T and T'.
(vii) Join PT and PT'. Then PT and PT' are the required tangents.

Steps of construction:
1) Construction a Δ ABC such that AB = 4 cm, AC = 6 cm, BC = 6 cm
2) Draw the bisectors of ∠ A and ∠ B and let them meet at O.
3) With O as centre draw OM the right bisector of AB meeting AB at D.
4) With O as centre and OD as radius draw a circle. The circle touches the sides of the Δ ABC at D, E, and F. Then this is the required incircle of Δ ABC.

Steps of construction:

1) Draw a circle with centre O and radius equal to 3 cm.
2) Draw a diameter AC
3) Draw another diameter BD which bisects AC at right ∠s.
4) Join AB, BC, CD and DA.
5) Now draw tangents to the given circle at the points A, B, C, D and let them meet at P, Q, R, S. Then PQRS is the required square about the given circle.

Steps of construction:

1) Draw an line segment BC = 5.2
2) With centre B and radius BA = 4.4 cm, draw an arc.
3) With C centre and radius CA = 7.1 cm, draw an arc intersecting the previous arc at A. Then ABC is the given triangle.
4) Draw the perpendicular bisectors of any two sides, say BC and AC, intersecting at O. Then O is the circumference of ΔABC.

Steps of construction:
1) Draw a line segment AC = 4 cm.
2) Draw ∠ CAX = 60°.
3) Draw the perpendicular bisector MN of AC.
4) Draw EA ⊥ AX at point A which intersects MN at O.
5) With Centre O and radius OA draw a circle.
6) Mark a point B on the circumference of the circle such that AB = 1.5 cm and mark a point D on the circumference so that AD = 2.0 cm.
7) Join BC and measure ∠ ABC = 60°. Then ABCD is the required cyclic quadrilateral.

Steps of construction:
(i) 1) Draw BC = 5 cm.
2) With B and C as centres draw two arcs to length 7 cm cutting each at A.
3) Join AB and AC
4) Then ABC is required triangle.
(ii) Draw AD, the right bisector of BC.
(iii) With A as centre and radius 3 cm draw a circle meeting AD in P.
(iv) 1) Join BP and CP.
2) Draw the right bisector of CP meeting AD in O.
3) With O as centre and radius equal to Op draw the required circle, passing through B and C.

Steps:
Draw a line segment AB = 5 cm long. Make an angle of 75° at 'B' draw perpendicular bisector of AB and the angular bisector of B.
Make 3.5 cm on the perpendicular bisector with O as centre and radius equal to OA or OB draw circumcircle. Mark 2.5 cm on AC from A. Join BD, it will intersect at P, with P as centre and PD as radius draw another circle.

Steps of construction:
(i) Draw a line segment BC = 6 cm.
(ii) At point B, make ∠ CBA = 55°, and at point C, make an angle ∠ BCA = 70° with the help of a protractor. Join AB and AC.
(iii) Draw the perpendicular bisectors of sides AB and AC. Let them intersects at O.
(iv) With O as centre and radius OA, draw a circle which passes through the point A, B and C. This is the required circumcircle of ΔABC.

Steps of constructions:
(i) Draw ∠ PQX = 90°
(ii) Bisects ∠ PQX and draw ∠ PQR = 45°
(iii) Cut off QS = 4.5 cm from QR.
(iv) r bisector of QS and it intersects of QX at O.
(v) With O as centre and radius = OQ ( or OS draw the required circle).

Steps of Construction:
i) Draw a line segment OP = 6 cm
ii) With centre O and radius 3.5 cm, draw a circle
iii) Draw the midpoint of OP
iv) With centre M and diameter OP, draw a circle which intersect the circle at T and S
v) Join PT and PS.
PT and PS are the required tangents. On measuring the length of PT = PS = 4.8 cm