Mathematics [4 marks sum]

   
 
Steps of Construction:
i) Draw a line segment BC = 5 cm
ii) With centers B and C, draw two arcs of 5 cm radius each which intersect each other at A.
iii) Join AB and AC.
iv) Draw angle bisectors of ∠B and ∠C intersecting each other at O.
v) From O, draw OL ⊥ BC.
vi) Now with centre O and radius OL, draw a circle which will touch the sides of ΔABC
On measuring, OL = 1.4 cm

 
Steps of construction:
i) Draw a line segment BC = 4.5 cm
ii) With centers B and C, draw two arcs of radius 4.5 cm which intersect each other at A.
iii) Join AC and AB.
iv) Draw perpendicular bisectors of AC and BC intersecting each other at O.
 v) With centre O, and radius OA or OB or OC draw a circle which will pass through A, B and C.
This is the required circumcircle of triangle ABC. On measuring the radius OA = 2.6 cm

  
Steps of Construction:
i) Draw a circle with centre O and radius BC = 4.5 cm
ii) Draw arcs making an angle of 180º - 60º = 120º at O such that ∠ AOB = 120º
iii)AT A and B, draw two rays making an angle of 90º at each point which meet each other at point P, outside the circle.
iv)AP and BP are the required tangents which make an angle of 60º with each other at P.

 
Steps of Construction:
i) Draw a circle with centre O and radius BC = 5 cm
ii) Draw arcs making an angle of 180º - 45º = 135º at O such that∠ AOB = 135º
iii) AT A and B, draw two rays making an angle of 90º at each point which meet each other at point P, outside the circle.
iv) AP and BP are the required tangents which make an angle of 45º with each other at P. 

 
i. Draw a circle of diameter 9 cm, taking O as the centre.
ii. Mark a point P outside the circle, such that PO = 7.5 cm.
iii. Taking OP as the diameter, draw a circle such that it cuts the earlier circle at A and B.
iv. Join PA and PB. 

Steps for construction :

1. Draw BC = 7.2 cm.
2. Draw an angle ABC = 90°using compass.
3. Draw BD perpendicular to AC using a compass.
4. Join BD.
5. Draw perpendicular bisectors of AB and BC which intersect atI, where I is the circumcentre of a circle.
6. Draw circumcircle using circumcentre I. we get the radius of a circle is 4.7 cm.

Steps for construction :

1. Draw concentric circles of radius 4 cm and 6 cm withcentre of O.
2. Take point P on the outer circle.
3. iii. Join OP.
4. Draw perpendicular bisectors of OP where M is the midpoint of OP.
5. Take a distance of a point O from the point M and mark arcs from M on the inner circle it cuts at point A and B respectively.
6. Join PA and PB.
7. We observe that PA and PB are tangents from outer circle to inner circle are equal of a length 4.5 cm each.

Steps for construction :

1. Draw BC = 6.8 cm.
2. Mark point D where BD = DC = 3.4 cm which is mid-point of BC.
3. iii. Mark a point A which is intersection of arcs AD = 4.4 cm and AB = 5 cm from a point D and B respectively.
4. Join AB, AD and AC.
5. ABC is the required triangle.
6. Draw bisectors of angle B and angle C which are ray BX and CY where I is theincentre of a circle.
7. Drawincircle of a triangle ABC

Steps of construction:
1) Draw AB=5.5 cm
2) construct ∠BAR=105° 
3) With centre A and redius 6 cm, cut off arc on AR at C. 
4) Join BC. ABC is the reqqiued triangle.
 (1) Draw angle bisector BD of  ∠ABC, which is the locus of points equidistant from BA and BC. 
(2) Draw perpendicular bisctor EF of BC, which is the loucs of point equidistant from B and C. 
(3) BD and EF intersect each other at point P.Thus, P satisfies the above two loci By Mesurment,PC=4.8 cm
 

 
Steps Of Construction:
i) Draw a circle with centre O and radius 3 cm.
ii) From O, take a point P such that OP = 5 cm
iii) Draw a bisector of OP which intersects OP at M.
iv) With centre M, and radius OM, draw a circle which intersects the given circle at A and B.
v) Join AP and BP.
AP and BP are the required tangents.
On measuring AP = BP = 4 cm 

 
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 

 
Steps of Construction:
(i) Draw a circle of radius 4 cm with centre O
(ii) Since the interior angle of regular hexagon is 60o, draw radii OA and OB such that ∠AOB= 60° .  
(iii) Cut off arcs BC, CD, EF and each equal to arc AB on given circle
(iv) Join AB, BC, CD, DE, EF, FA to get required regular hexagon ABCDEF in a given circle. 
The circle is the required circum circle, circumscribing the hexagon.     

  
Steps of Construction:
i) Draw a line segment AB = 5.8 cm
ii) At A and B, draw rays making an angle of 120o each and cut off AF = BC = 5.8 cm
iii) Again F and C, draw rays making an angle of 120o each and cut off FE = CD = 5.8 cm.
iv) Join DE. Then ABCDEF is the regular hexagon.
v) Draw the bisectors of ∠A and ∠B intersecting each other at O.
vi) From O, draw OL ⊥ AB 
vii) With centre O and radius OL, draw a circle which touches the sides of the hexagon.
This is the required in circle of the hexagon.
 

  
Steps of Construction:
i) Draw a line segment BC = 5.6 cm
ii) With centers B and C, draw two arcs of 5.6 cm radius each which intersect each other at A.
iii) Join AB and AC.
iv) Draw angle bisectors of ∠B and ∠C intersecting each other at O.
v) From O, draw OL ⊥ BC. 
vi) Now with centre O and radius OL, draw a circle which will touch the sides of ΔABC. 
This is the required circle.

 
Steps of construction:
i) Draw a line segment BC = 6 cm
ii) With centers B and C, draw two arcs of radius 6 cm which intersect each other at A.
iii) Join AC and AB.
iv) Draw perpendicular bisectors of AC, AB and BC intersecting each other at O.
v) With centre O, and radius OA or OB or OC draw a circle which will pass through A, B and C. This is the required circumcircle of triangle ABC.