Exercise 14 G
Question 1
Ans: (i) Draw a circle with center O and A suitable radius.
(ii)Take of point $P$ outside the circle.
(iii) Join OP and take its midpoint $M$.
(iv) with center mand diameter op, draw a circle which intersects the given circle at T and S
(V) Join PT and PS.
PT and PS are required tangents to the circle on measuring $P T=P S=5.5 \mathrm{~cm}$
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Question 2
Ans: (i) Draw a circle with center O and with a suitable radius
(ii) Take a point P on it and Join OP .
(iii) At P draw a perpendicular to op which meet a given line at S
Then ST is the required tangent.
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Question 3
Ans: (i) Draw a circle with center O and radius $4 \mathrm{~cm}$.
(ii) Take a point P Such that OP= 5cm
(iii)Draw its bisector which bisects OP =5cm
(iv) With center M and radius MP draw a circle intersecting the given circle at T and S.
(v) Join PT and PS.
PT and PS are the required tangents to the circle on Measuring each of them PT=PS=3cm
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Question 4
Ans: (i) Draw a circle with center O and radius 2cm
(ii) Take a point P outside the circle
(iii) From P draw a straight line which intersects the circle at A and B
(iv) With BP as diameter draw a semicircle
(v) At A, draw a perpendicular which meets the semicircle at c
(vi) With center P and radius PC, draw an arc which intersects the given circle at T and S.
(vii) Join PT.
PT is the required tangent.
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Question 5
Ans:( i) Draw a circle with center $O$ and some suitable radius.
(ii) Take a point P on it.
(iii) Take two more points $Q$ and $R$ on the remaining part of the circle and Joined $P Q, Q R$ and $R P$.
(iv) Draw an angle $\angle Q P T$ equal to $\angle R$ and Produce the line TP to S.
Then SPT is the required tangent to the circle
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Question 6
Ans: (i) Draw a circle with center O and a suitable radius.
(ii) Draw a radius OS and on PS, draw an angle $\angle S O T$ of $180^{\circ}-60^{\circ}=120^{\circ}$
(iii) At S and T draw lines making $90^{\circ}$ each. which intersect each other at P
Then PT and SP are the required tangents making an angle of 60 with each other at P on measuring them each one of them is 4.5cm
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Calculation :
Here radius of the circle (r)= 3cm and distance of OP= 5.3cm
In right $\triangle O P T$,
$P T^{2}=O P^{2}-O T^{2}=(5.3)^{2}-(3)^{2}$
$=28.09-9=$ (9.09)= $(4.37)^{2}$
So $P T=4.37 \mathrm{~cm}$
Question 7
Ans: (i) Draw a line segment $A C=7 \mathrm{~cm}(b=7 \mathrm{~cm})$
(ii) At $C$, draw a ray $C X$ making an angle of $30^{\circ}$
(iii) With center A and Radius 6cm(c= 6cm) Draw are which intersects CX at B and B'
(iv) Join AB and AB'
Then two triangle are possible $\triangle A B C$ and $\triangle A B^{\prime} C$ in which a= 1.3cm or 11.3cm
(v) Now draw the perpendicular bisector of AC and BC which intersect each other at O.
(vi) With center O and radius equal to OB, Draw a circle which passes through A,B and C,
Then this is the required circumcircle of the $\triangle A B C$. on measuring its radius $=6.5 \mathrm{~cm}$
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Question 8
Ans: (i) Draw a line segment BC = 6cm
(ii) At B, draw a ray BX making an angle of $90^{\circ}$ and cut off $B A=4 \mathrm{~cm}$
(iii) Join AC.
(iv) Now draw the perpendicular bisects of AB and DC intersecting each other at O.
(V) With center O and radius OA, draw a circle which will pass through A,B and C.
This is required circumcircle of $\triangle A B C$ Whose radiues is $\frac{1}{2} A C=3.6 \mathrm{~cm}$
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Question 9
Ans: Steps of constructions:
(i) Draw a line segment $B C=4 \mathrm{~cm}$
(ii) At C, draw a ray CX making a angle of 45 and CY making an angle of 90
(iii) Cut off CQ = 2.5cm
(iv) From Q, draw a line PQ parallel to BC. Which meets CX at A.
(v) Join A B
(vi) Draw the perpendicular bisects of AB and BC which intersects each other at O.
(vii) With center O and radius OA, draw a circle which will pass through A,B and C.
This is the required circumcircle of $\triangle A B C$ Whose radius OA= 2.1cm
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Question 10
Ans: It is true
Question 11
Ans: (i) Draw a line segment BC = 4cm
(ii) With center B and C and Radius 4cm , draw arcs which intersect each other at A.
(iii) Join AB and CA .
Then $\triangle A B C$ is an equilateral triangle
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(iv) Draw the perpendicular bisects of sides AB and BC which intersect each other at O.
(v) with center $O$ and radius OA, draw a circle which will pass through A,B, C .
Question 12
Ans: Steps of construction:
(i) Draw the given $\triangle A B C$.
(ii) Draw the angle bisects of $\angle B$ and $\angle C$ which intersects each other at I.
(iii) From I . Draw a perpendicular ID and BC.
(iv) with Center I and radius ID, draw a circle which touches the sides of the triangle ABC at D,E and F.
On measuring the radius ID=1.4cm
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Question 13
Ans: (i)Draw the line segment BC =5cm
(ii) with centers $B$ and $C$ and radius $5 \mathrm{~cm}$, draw arcs intersecting each other at $A$.
(iii) Join $A P$ and $A C$.
$\triangle A B C$ is an equilateral triangle.
(iv) Draw the angle bisectors of $\angle B$ and $\angle C$ which intersect each other at 1 .
(v) From 1. draw ID $\perp B C .$
(vi) With center I and radius ID draw a circle which will the sides of $\triangle A B C$ at $D E$ and $F$.
On measuring the radius ID = 1.5cm
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Question 14
Ans: (a) (i) Draw a line segment AB = 8cm
(ii) With center A and Radius 5cm and With center B and Radius 6cm Draw arcs Which intersect each other At C.
(iii) Join $A C$ and $B C$.
$\triangle A B C$ is the required mangle.
(b) (i) Draw the bisector of $\angle A$ and $\angle B$ Which intersect each other at I. I is the incenter of the in circle
(c) (ii) From L, cut off $L P=L Q=1 \mathrm{~cm}$ So that PQ = 2cm
(iii) With center I and Radius IP Draw a circle which intersects BC at R and S CA at T and U.
Then chords PQ= RS= TU =2cm.
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Question 15
Ans: ( i) Draw a line segment $A B=3.2 \mathrm{~cm}$.
(ii) Draw rays at $A$ and $B$ making angle of $120^{\circ}$ each and cut off $A F=B C=3.2 \mathrm{~cm}$
(iii) Similarly at F and C, draw rays making angle of 120 each and cut off FE =CD = 3.2cm
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(iv) Join ED
$A B C D E f$ is a regular hexagon.
(v) Draw the angle bisects of $\angle A$ and $\angle B$ which intersect each other at O .
(vi) From O, draw QL⊥ AB
(vii) with center $O$ and radius $O L$ draw a circle which will touch the sides regular of regular hexagon ABCDEF.
Question 16
Ans: (i) Draw a line segment $A B=2.8 \mathrm{~cm}$
(ii) At $A$ and $B$ draw rays making angle of $120^{\circ}$ each and cut off $A F=B C=2.8 \mathrm{~cm}$
(iii) similarly at $F$ and $C$,draw rays making angle of $120^{\circ}$ and we Cut $F E=C D=2.8 \mathrm{~cm}$
(iv) Join ED.
ABCDEF is a regular hexagon.
(v) Draw the perpendicular bisects of $A B$ and $A F$ which intersect each other at O.
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