# Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 19 - Constructions (Circles)

## Selina Textbook Solutions Chapter 19 - Constructions (Circles)

Selina Textbook Solutions are a perfect way to ace your examination with high marks. These Textbook Solutions are extremely helpful for solving difficult questions in the ICSE Class 10 Mathematics exam. Our Selina Textbook Solutions are written by our subject experts. Find all the answers to the Selina textbook questions of** **Chapter 19 - Constructions (Circles).

All Selina textbook questions of** **Chapter 19 - Constructions (Circles) solutions are created in accordance with the latest ICSE syllabus. These free Textbook Solutions for ICSE Class 10 Selina Concise Mathematics will give you a deeper insight on the fundamentals in this chapter and will help you to score more marks in the final examination. ICSE Class 10 students can refer to these solutions while doing their homework and while studying and revising for the Mathematics exam.

## Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 19 - Constructions (Circles) Page/Excercise 19

**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

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

**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.

**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 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) Construction of triangle:

a) Draw a line segment BC = 7 cm

b)
At B, draw a ray BX making an angle of 45^{o}_{ }and cut off
BE = AB - AC = 1 cm

c) Join EC and draw the perpendicular bisector of EC intersecting BX at A.

d) Join AC.

_{} is the required triangle.

ii) Construction of incircle:

e)
Draw angle bisectors of _{}and _{}intersecting
each other at O.

f) From O, draw perpendiculars OL to BC.

g)
O as centre and OL as radius draw circle which touches the sides of the _{}.
This is the required in-circle of _{}.

On measuring, radius OL = 1.8 cm

**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 _{}and _{} intersecting
each other at O.

v)
From O, draw _{}.

vi)Now
with centre O and radius OL, draw a circle which will touch the sides of _{}

On measuring, OL = 1.4 cm

**Steps
of construction:**

i) Draw a line segment AB = 6 cm

ii)
At A, draw a ray making an angle of 60^{o}_{ }with BC.

iii) With B as centre and radius = 6.2 cm draw an arc which intersects AX ray at C.

iv) Join BC.

_{} is the required triangle.

v) Draw the perpendicular bisectors of AB and AC intersecting each other at O.

vi) With centre O, and radius as OA or OB or OC, draw a circle which will pass through A, B and C.

vii)
From O, draw _{}.

Proof:
In right _{}and _{}

OA = OB (radii of same circle)

Side OD = OD (common)

_{}

**Steps of Construction:**

i)Draw a line segment BC = 4 cm.

ii) At C, draw a perpendicular line CX and from it, cut off CE = 2.5 cm.

iii) From E, draw another perpendicular line EY.

iv) From C, draw a ray making an angle of 45^{o} with CB, which intersects EY at A.

v) Join AB.

_{} is the required triangle.

vi) Draw perpendicular bisectors of sides AB and BC intersecting each other at O.

vii) With centre O, and radius OB, draw a circle which will pass through A, B and C.

Measuring the radius OB = OC = OA = 2 cm

i)
O is called the circumcentre of circumcircle of _{}.

ii) OA, OB and OC are the radii of the circumcircle.

iii) Yes, the perpendicular bisector of BC will pass through O.

i)
O is called the incentre of the incircle of _{}.

ii) OR and OQ are the radii of the incircle and OR = OQ.

iii) OC is the bisector of angle C

_{}

**Steps of Construction:**

i) Draw a line segment BC = 6 cm.

ii) With centre B and radius 8 cm draw an arc.

iii) With centre C and radius 5 cm draw another arc which intersects the first arc at A.

iv) Join AB and AC.

_{} is the required triangle.

v) Draw the angle bisectors of _{}intersecting each other at I. Then I is the incentre of the triangle ABC

vi) Through I, draw _{}

vii) Now from D, cut off _{}

viii) With centre I, and radius IP or IQ, draw a circle which will intersect each side of triangle ABC cutting chords of 2 cm each.

**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.

**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 _{}and _{} intersecting
each other at O.

v)
From O, draw _{}.

vi)Now
with centre O and radius OL, draw a circle which will touch the sides of _{}

This is the required circle.

**Steps of Construction:**

i) Draw a regular hexagon ABCDEF with each side equal to 5 cm and each interior angle 120º.

ii) Join its diagonals AD, BE and CF intersecting each other at O.

iii) With centre as O and radius OA, draw a circle which will pass through the vertices A, B, C, D, E and F.

This is the required circumcircle.

**Steps of Construction:**

i) Draw a line segment AB = 5.8 cm

ii) At A and B, draw rays making an angle of 120^{o} each and cut off AF = BC = 5.8 cm

iii) Again F and C, draw rays making an angle of 120^{o} each and cut off FE = CD = 5.8 cm.

iv) Join DE. Then ABCDEF is the regular hexagon.

v) Draw the bisectors of _{} intersecting each other at O.

vi) From O, draw _{}

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 circle of radius 4 cm with centre O

(ii)
Since the
interior angle of regular hexagon is 60^{o}, draw radii OA and OB
such that _{}

(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 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

i.

- Draw a line BC = 5.4 cm.
- Draw AB = 6 cm, such that m∠ABC = 120°.
- Construct the perpendicular bisectors of AB and BC, such that they intersect at O.
- Draw a circle with O as the radius.

ii.

(e) Extend the perpendicular bisector of BC, such that

it intersects the circle at D.

(f) Join BD and CD.

(g) Here BD = DC.

Steps of construction:

- Draw AF measuring 5 cm using a ruler.
- With A as the centre and radius equal to AF, draw an arc above AF.
- With F as the centre, and same radius cut the previous arc at Z
- With Z as the centre, and same radius draw a circle passing through A and F.
- With A as the centre and same radius, draw an arc to cut the circle above AF at B.
- With B as the centre and same radius, draw an arc to cut the circle at C.
- Repeat this process to get remaining vertices of the hexagon at D and E.
- Join consecutive arcs on the circle to form the hexagon.
- Draw the perpendicular bisectors of AF, FE and DE.
- Extend the bisectors of AF, FE and DE to meet CD, BC and AB at X, L and O respectively.
- Join AD, CF and EB.

These are the 6 lines of symmetry of the regular hexagon.

Steps for construction:

- Draw AB = 5 cm using a ruler.
- With A as the centre cut an arc of 3 cm on AB to obtain C.
- With A as the centre and radius 2.5 cm, draw an arc above AB.
- With same radius, and C as the centre draw an arc to cut the previous arc and mark the intersection as O.
- With O as the centre and radius 2.5 cm, draw a circle so that points A and C lie on the circle formed.
- Join OB.
- Draw the perpendicular bisector of OB to obtain the mid-point of OB, M.
- With the M as the centre and radius equal to OM, draw a circle to cut the previous circle at points P and Q.
- Join PB and QB. PB and QB are the required tangents to the given circle from exterior point B.

QB = PB = 3 cm

That is, length of each tangent is 3 cm.

## Selina Concise Mathematics X Class 10 Chapter Solutions

- Chapter 1 - Value Added Tax
- Chapter 2 - Banking (Recurring Deposit Accounts)
- Chapter 3 - Shares and Dividends
- Chapter 4 - Linear Inequations (in one variable)
- Chapter 5 - Quadratic Equations
- Chapter 6 - Solving (simple) Problmes (Based on Quadratic Equations)
- Chapter 7 - Ratio and Proportion (Including Properties and Uses)
- Chapter 8 - Remainder And Factor Theorems
- Chapter 9 - Matrices
- Chapter 10 - Arithmetic Progression
- Chapter 11 - Geometric Progression
- Chapter 12 - Reflection (In x-axis, y-axis, x=a, y=a and the origin ; Invariant Points)
- Chapter 13 - Section and Mid-Point Formula
- Chapter 14 - Equation of a Line
- Chapter 15 - Similarity (With Applications to Maps and Models)
- Chapter 16 - Loci (Locus and its Constructions)
- Chapter 17 - Circles
- Chapter 18 - Tangents and Intersecting Chords
- Chapter 19 - Constructions (Circles)
- Chapter 20 - Cylinder, Cone and Sphere (Surface Area and Volume)
- Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables)
- Chapter 22 - Heights and Distances
- Chapter 23 - Graphical Representation (Histograms and Ogives)
- Chapter 24 - Measures of Central Tendency (Mean, Median, Quartiles and Mode)
- Chapter 25 - Probability

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