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Page No 114:

Question 1:

Choose the correct answer from the given four options:
To divide a line segment AB in the ratio 5:7, first a ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is
(A) 8
(B) 10
(C) 11
(D) 12

Answer:

Given, a line segment AB in the ratio 5 : 7.
A : B = 5 : 7
Now,
Draw a ray AX which makes an acute angle ∠BAX, then mark A + B points at equal distance. Here, A = 5 and B = 7.

Therefore, the minimum number of these points
= A + B
= 5 + 7
= 12.
Hence, the correct answer is option D.

Page No 114:

Question 2:

Choose the correct answer from the given four options:
To divide a line segment AB in the ratio 4:7, a ray AX is drawn first such that ∠BAX is an acute angle and then points A1, A2, A3, .... are located at equal distances on the ray AX and the point B is joined to
(A) A12
(B) A11
(C) A10
(D) A9

Answer:

Given, a line segment AB in the ratio 4 : 7
A : B = 4 : 7
Minimum number of points located at equal distances on the ray AX
= A + B
= 4 + 7
= 11
A1, A2, A3, .......... are located at equal distances on the ray AX.
Point B is joined to the last point is A11.
Hence, the correct answer is option B.
 

Page No 114:

Question 3:

Choose the correct answer from the given four options:
To divide a line segment AB in the ratio 5 : 6, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY parallel to AX and the points A1, A2, A3, ... and B1, B2, B3, ... are located at equal distances on ray AX and BY, respectively. Then the points joined are
(A) A5 and B6
(B) A6 and B5
(C) A4 and B5
(D) A5 and B4

Answer:

Given, a line segment AB in the ratio 5 : 7
A : B = 5 : 7

Steps of construction:
1. Draw a ray AX, an acute BAX.

2. Draw a ray BYAX,
ABY = BAX.

3. Now, locate the points A1, A2, A3, A4 and A5 on AX and B1, B2, B3, B4, B5 and B6 on BY (Because A : B = 5 : 6)

4. Join A5B6.
 A5B6 intersect AB at a point C.
AC : BC = 5 : 6

Hence, the correct answer is option A.

 

Page No 114:

Question 4:

Choose the correct answer from the given four options:
To construct a triangle similar to a given ∆ABC with its sides 37 of the corresponding sides of ∆ABC, first draw a ray BX such that ∠CBX is an acute angle and X lies on the opposite side of A with respect to BC. Then locate points B1, B2, B3, ... on BX at equal distances and next step is to join
(A) B10 to C
(B) B3 to C
(C) B7 to C
(D) B4 to C

Answer:

Steps:
1. Locate points B1, B2, B3, B4, B5, B6 and B7 on BX at equal distance.


2. Join the last points is B7 to C.


Hence, the correct answer is option C.

Page No 114:

Question 5:

Choose the correct answer from the given four options:
To construct a triangle similar to a given ∆ABC with its sides 85 of the corresponding sides of ∆ABC draw a ray BX such that ∠CBX is an acute angle and X is on the opposite side of A with respect to BC. The minimum number of points to be located at equal distances on ray BX is
(A) 5
(B) 8
(C) 13
(D) 3

Answer:

To construct a triangle similar to a triangle, with its sides mn of the corresponding sides of given triangle, the minimum number of points to be located at an equal distance is equal to the greater of m and n in mn.
Here, mn=85

So, the minimum number of point to be located at equal distance on ray BX is 8.
Hence, the correct answer is option C.

Page No 114:

Question 6:

Choose the correct answer from the given four options:
To draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be
(A) 135°
(B) 90°
(C) 60°
(D) 120°

Answer:


In the given figure,
OP and OQ are the tangents to the circle.
Now,
Sum of opposite angles = 180°
POQ + PRQ = 180°
⇒ 60° + θ = 180°
⇒ θ = 120°
The angle between them should be 120°.
Hence, the correct answer is option D.



Page No 115:

Question 1:

Write True or False and give reasons for your answer
By geometrical construction, it is possible to divide a line segment in the ratio 3:13.

Answer:

Given, ratio =  3:13
   Required ratio = 3:1        [ multiplying 3 in each term]
So, 3:13 can be simplified as 3 : 1 and 3 as well as 1 both are positive integers.
Hence, by geometrical construction is possible to divide a line segment in the ratio 3 : 1.

Page No 115:

Question 2:

Write True or False and give reasons for your answer
To construct a triangle similar to a given ∆ABC with its sides 73of the corresponding sides of ∆ABC, draw a ray BX making acute angle with BC and X lies on the opposite side of A with respect to BC. The points B1, B2, ...., B7 are located at equal distances on BX, B3 is joined to C and then a line segment B6C' is drawn parallel to B3C where C' lies on BC produced. Finally, line segment A'C' is drawn parallel to AC.

Answer:

False

Steps of construction
1. Draw a line segment BC.
2. B and C as centers draw two arcs of suitable radius intersecting each other at A.
3. Join BA and CA. ∆ABC is required triangle.
4. From B draw any ray BX downwards making an acute angle CBX.
5. Marked seven points B1, B2, B,............B7 on BX (BB1 = B1B2 = B2B3 = B3B4 = B4B5 = B5B6 = B6B7).
6. Join B3C and from B7 draw a line B7C’  B3C intersecting the extended line segment BC at C’.
7. Draw C’A’  CA intersecting the extended line segment BA at A’.
Then, ∆A’BC’ is the required triangle whose sides are 73 of the corresponding sides of ∆ABC.

Given that, segment B6C’ B3C. But since our construction is never possible that segment B6C’  B3C because the similar triangle A’BC’ has its sides 73 of the corresponding sides of triangle ABC.
So, B7C’ is parallel to B3C.

Page No 115:

Question 3:

Write True or False and give reasons for your answer
A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of 3 cm from the centre.

Answer:

False

Let, r = radius of circle and d = distance of a point from the center.
r = 3.5 and d = 3
 r > d
Point P lies in inside the circle, as shown below:

Page No 115:

Question 4:

Write True or False and give reasons for your answer
A pair of tangents can be constructed to a circle inclined at an angle of 170°.

Answer:

True
If the angle between the pair of tangents is always > 0° or <180 °,
Then we can construct a pair of tangents to a circle.
Hence, we can draw a pair of tangents to a circle inclined at an angle of 170°, as shown below:



Page No 116:

Question 1:

Draw a line segment of length 7 cm. Find a point P on it which divides it in the ratio 3 : 5.

Answer:

Steps of construction:
1. Draw a line segment AB = 7 cm.



2. Draw a ray AX, making an acute ∠BAX.



3. Along AX, mark 3 + 5 = 8 points A1, A2, A3, A4, A5, A6, A7, A8 such that AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7 = A7A8.


4. Join A8B.


5. From A3, draw A3 A8B meeting AB at P. (By making an angle equal to ∠BA8 at A3).
Then P is the point on AB which divides it in the ratio 3 : 5. Thus, AP : PB = 3 : 5.


Justification:
Let
AA1 = A1A2 = A2A3 = A3A4 =........= A7A8 = x
In ∆ABA8, A3P   A8B

APPB=AA3A3A8=3x5x=35
Hence, AP : PB = 3 : 5.

 

Page No 116:

Question 2:

Draw a right triangle ABC in which BC = 12 cm, AB = 5 cm and ∠B = 90°.
Construct a triangle similar to it and of scale factor 23. Is the new triangle also a right triangle?

Answer:

Steps of construction:
1. Draw a line segment BC = 12 cm.



2. From B draw a line which makes a right angle.

a. Now as point B is the initial point as the centre, draw an arc any radius such that, the arc meets the ray BC at point D.



b. With D as centre and with the same radius as before, draw another arc cutting the previous one at point E.



c. Now with E as centre and with the same radius, draw an arc cutting the first arc (drawn in step a) at point F.



d. With E and F as centres, and with a radius more than half the length of FE, draw two arcs intersecting at point G.



e. Join points B and G. The angle formed by GBC is 90°. i.e. ∠ GBC = 90°.



3. From B as centre draw an arc of 5 cm which intersects the line GB at A.



4. Join AC, ABC is the given right triangle.



5. From B draw an acute CBH downwards.



6. On ray BH, mark three point B1,B2 and B3, such that BB1 = B1B2 = B2B3.



7. Join B3C



8. From point B2 draw B2N || B3C intersect BC at N.



9. From point N draw NM | | CA intersect BA at M. ∆MBN is the required triangle. ∆MNB is also a right angled triangle at B.



Justification:
As per the construction, ∆MNB is the required triangle
Let BB1 = B1B2 = B2B3 = x

Considering ∆BNB2 and ∆BCB3
∠ B = ∠ B                                      [same]
∠BNB2 =∠BCB                          [corresponding angles of the same transverse as B3C || B2N]
BNB2 ~BCB                     [ by AA criteria ]

 BNBC=BB2BB3=2x3x=23                      .....(1)

Similarly,
In ∆MBN and ∆ABC ,
∠ A  = ∠ A                                   [same]
∠BAC = ∠BMN                          [corresponding angles of the same transverse as AC || MN]
∆MBN ~ ∆ABC                           [ by AA criteria ]

MBAB=MNAC=BNBC=23                      .....(2)

Constructed triangle ∆MNB is of scale 23 times of the ∆ABC.

Also, as we can clearly see, ∠B = 90° is common in both the triangles ∆ABC and ∆MNB. Hence the constructed triangle ∆MNB is also a right angle triangle.


 

Page No 116:

Question 3:

Draw a triangle ABC in which BC = 6 cm, CA = 5 cm and AB = 4 cm. Construct a triangle similar to it and of scale factor 53.

Answer:

Steps of construction:

1. Draw the line segment BC = 6 cm.




2. Taking B and C as centres, draw two arcs of radii 4 cm and 5 cm respectively intersecting each other at A.



3. Join BA and CA. ∆ABC is the required triangle.



4. From B, draw any ray BD downwards making at acute angle.



5. Mark five points B1, B2, B3, B4 and B5 on BD, such that
BB= B1B2 = B2B3 = B3B4 = B4B5


6. Join B3C and from B5 draw B5M || B3C intersecting the extended line segment BC at M.


7. From point M draw MN || CA intersecting the extended line segment BA at N.



8. Then, ∆NBM is the required triangle whose sides are equal to 53 of the corresponding sides of the ∆ABC.

Justification:
As per the construction, ∆MNB is the required triangle
Let
BB1 = B1B2 = B2B3 = B3B4 = B4B5 = x

In triangles ∆BCB3 and ∆BMB5
∠ B = ∠ B                                   [same angle]
∠BCB3 = ∠BMB5        [corresponding angles of the same transverse as B5D || B3C]
 ∆BCB3 ~ ∆BMB5                        [by AA congruency criteria]

BMBC=BB5BB3=5x3x=53                           .....(1)

Similarly, in ∆MBN and ∆CBA 

∠ B = ∠ B                                          [same angle]
∠BAC = ∠BNM                          [corresponding angles of the same transverse asAC || MN]
∆MBN ~ ∆CBA                            [by AA congruency criteria]

ABNB=ACNM=BMBC=53                           .....(2)

From (1) and (2) the constructed triangle ∆NBM is of scale 53 times of the ∆ABC.

 

Page No 116:

Question 4:

Construct a tangent to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre.

Answer:

Steps of construction

1. Draw a circle of radius 4 cm. Let centre of this circle be O.



2. Take a point M at 6 cm away from the radius.



3. Join OM and bisect it. Now, with M and O as centres and with radius more than half of draw two arcs on the either sides of the line OM. Let the arc meet at A and B just that, M1 be mid-point of OM.



4. Taking M1 as centre and M1O as radius draw a circle to intersect circle with radius 4 and centre O at two points P and Q.


5. Join PM and QM. PM and QM are the required tangents from M to circle with centre O and radius 4.


 



Page No 117:

Question 1:

Two line segments AB and AC include an angle of 60° where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that AP=34 AB and AQ=14 AC. Join P and Q and measure the length PQ.

Answer:

Thinking process:
I. Firstly, we find the ratio of AB in which P divides it with the help of the relation AP = 34 AB.
II. Secondly, we find the ratio of AC in which Q divides it with the help of the relation AQ = 14 AC.
III. Now, construct the line segment AB and AC in which P and Q respectively divides it in the ratio from step
(i) and (ii) respectively.
IV. Finally get the points P and Q. After that join PQ and get the required measurement of PQ.

Given that, AB = 5 cm and AC = 7 cm
Also, AP=34AB   and AQ=14AC                        .....(1)
AP=34×5 = 154 cm
Then, PB =AB - AP
PB=5-154 =54cm                   
AP : PB=154 : 54
Hence AP : PB = 3 : 1
i.e. scale factor of line segment AB is 31.
Again from (1)
AQ=14AC=14×7=74 cm
Then,
QC=AC-AQ=7-74=21 4cm
AQ : QC= 74 : 214
Hence AQ : QC = 1 : 3
i.e. scale factor of line segment AQ is 13.

Steps of construction:
1. Draw a line segment AB = 5cm.



2. Now draw a ray AZ making an acute ∠BAZ=60°.



3. With A as centre and radius equal to 7 cm draw an arc intersecting the line AZ at C.


4. Draw a ray AX, making an acute ∠BAX.



5. Along AX, mark 1+3 = 4 points A1, A2, A3 and A4
Such that A1A2 = A1A3 = A3A4


6. Join A4B


7. From A3 draw A3P || A4B meeting AB at P. [by making an angle equal to ∠AA4B]
Then, P is the point on AB which divides it in the ratio 3 : 1.
So, AP : PB = 3 : 1

8. Draw a ray AY, making an acute ∠CAY.



9. Along AY, mark 3+1 = 4 point B1, B2, B3 and B4.
Such that AB1 = B1B2 = B2B3 = B3B4


10. Join B4C.


11. From B1 draw B1Q || B4C meeting AC at Q.                                      [by making an angle equal to∠AB4C]
Then, Q is the point on AC which divides it in the ratio 1 : 3.
So, AQ : QC = 1 : 3


12. Finally, join PQ and its measurement is 3.25 cm.

 

Page No 117:

Question 2:

Draw a parallelogram ABCD in which BC = 5 cm, AB = 3 cm and ∠ABC = 60°, divide it into triangles BCD and ABD by the diagonal BD.
Construct the triangle BD' C' similar to ∆BDC with scale factor 43. Draw the line segment D'A' parallel to DA where A' lies on extended side BA. Is A'BC'D' a parallelogram?

Answer:

Thinking process:
I. Firstly we draw a line segment, then either of one end of the line segment with length 5 cm and making an angle 60° with this end. We know that is parallelogram both opposite sides are equal and parallel, then again draw a line with 5 cm making an angle with 60° from other end of line segment. Now, join both parallel line by a line segment whose measurement is 3 cm, we get a parallelogram. After that we draw a diagonal and get a triangle BOC.
II. Now, we construct ∆BD'C' similar to ∆BDC with scale factor 43.
III. Now, draw the line segment D'A' parallel to DA.
IV. Finally, we get the required parallelogram A'BC'D'.

Steps of construction:
1. Draw a line segment AB = 3 cm.



2. Now, draw a ray BY making an acute ∠ABY = 60°.



3. With B as centre and radius equal to 5 cm draw an arc cut the point C on BY.



4. Again draw a ray AZ making an acute ∠ZAX1 = 60°.


5. With A as centre and radius equal to 5 cm draw an arc cut the point D on AZ.



6. Now, join CD and finally make a parallelogram ABCD.



7. Join BD, which is a diagonal of parallelogram ABCD and from B draw any ray BX downwards making an acute ∠CBX.


8. Locate 4 points B1, B2, B3, B4 on BX, such that BB1 = B1B2 = B2B3 = B3B4.


9. Join B3C and from B4 draw a line B4C' || B3C intersecting the extended line segment BC at C'.


11. From point C' draw C'D'CD intersecting the extended line segment BD at D'. Then, ∆D'BC' is the required triangle whose sides are 43 of the corresponding sides of ∆DBC.

12. Now draw a line segment D'A' parallel to DA, where A' lies on the extended side BA i.e. a ray BX1.


13. Finally, we observe that A'BC'D' is a parallelogram in which A'D' = 6.5 cm A'B = 4 cm and ∠A'BD' = 60° divide it into triangles BC'D' and A'BD' by the diagonal BD'.



Justification that A'BC'D' is a parallelogram:
As per the construction,


CD || C'D' and BC' || A'D'                                                                           .....(1)
Also,
As per the given data, ∠ ABC = 60°.
Consider the parallelogram ABCD, the sum of complementary angles is 180°.
So ∠ ABC + ∠ BCD = 180°
∠ BCD = 180° - ∠ ABC = 180° - 60° = 120°.
Therefore ∠ BCD = 120°.

Now, as CD || C'D',
∠ BCD = ∠ BC'D'                                                                                                       (as corresponding angles)
Hence ∠ BC'D' = 120°                                                                               .....(2)

As BC || AD and AD || A'D', hence BC || A'D',
∠ABC = ∠D'A'X1                                                                                                        (corresponding angles)
So ∠D'A'X1 = ∠ABC = 60°
Now consider ∠D'A'X1 + ∠ D'A'B = 180°                                                                     (linear angle)
60° + ∠ D’A’B = 180°
∠ D’A’B = 180° - 60° = 120°                                                               .....(3)

From (1), (2) and (3), we can say that A’BC'D' is a parallelogram.                                (As opposite sides and opposite angles are equal).

Page No 117:

Question 3:

Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

Answer:

Given, two concentric circles of radii 3 cm and 5 cm with centre O. We have to draw pair of tangents from point P on outer circle to the other.
Steps of construction:
1. Draw two concentric circles with centre O and radii 3cm and 5cm.



2. Taking any point P on outer circle. Join OP.



3. Bisect OP, let M’ be the mid-point of OP.



4. Taking M’ as centre and OM’ as radius draw a circle dotted which cuts the inner circle at A and B.


5. Join PA and PB. Thus, PA and PB are required tangents.



6. On measuring PA and PB, we find that PA = PB = 4 cm.



Actual calculation:
In the right angle ∆OAP,
∠PAO = 90°
According to Pythagoras theorem
(hypotenuse)2 = (base)2 + (perpendicular)2
PA2 = (5)2 - (3)2 = 25 - 9 = 16
PA = 4 cm
Hence, the length of both tangents is 4 cm.
Therefore, PA = PB = 4 cm.

Page No 117:

Question 4:

Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5 cm.
Construct a triangle PQR similar to ∆ABC in which PQ = 8 cm. Also justify the construction.

Answer:

Steps of construction:
1. Draw a line segment BC = 5 cm.



2. Construct OQ the perpendicular bisector of line segment BC meeting BC at P'.



3. Taking B and C as centres draw two arcs of equal radius 6 cm intersecting each other at A.

4. Join BA and CA. So, ∆ABC is the required isosceles triangle and from B, draw any ray BX making an acute angle, ∠CBX.


5. Locate four points B1, B2, B3 and B4 on BX such that BB1 = B1B2 = B2B3 = B3B4

6. Join B3C and from B4 draw a line B4R || B3C intersecting the extended line segment BC at R.

7. From point R, draw RP || CA meeting the extended line BA at P.


Then, ∆PBR is the required triangles.

Justification:
Let BB1 = B1B2 = B2B3 = B3B4 = x
As per the construction B4B3C

BCCR=BB3B3B4=3xx=31
BCCR=31

BRBC=BC+CRBC=BCBC+CRBC=1+13=43

Also, from the construction RP || CA

In PBR and ABC,
PBR = ABC
PRB=ACB                                            [Corresponding Angles]
PBR~ABC                                        [ By AA criteria]
and PBAB=RPCA=BRBC=43

Hence, the new triangle is similar to the given triangle whose sides are 43 times of the corresponding sides of the isosceles ABC.

Page No 117:

Question 5:

Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60º. Construct a triangle similar to ∆ABC with scale factor 57. Justify the construction.

Answer:

Steps of construction:
1. Draw a line segment AB = 5 cm.
 



Page No 118:

Question 6:

Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is 60º. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

Answer:

Step - I  A circle with center O and radius 4cm is drawn.
Step - II  OQ is joined when Q is any point on the circle.
Step - III ∠OQR = 30° is drawn and QR intersects the circle at R.
Step - IV Centering Q and R and with radius QR two arcs are drawn, they intersect at P.
Step - V PQ and PR are joined.
PQ and PR are the required tangents.



Justification
PQ = QR = RP (By construction)
i.e. PQR=QRP=RPQ=60°OQP=OQR+PQR                 =30°+60°                 =90°
Thus, PQ is a tangent at Q.
Again in ∆OQR, we have
OQR=ORQ=30°
ORP=ORQ+QRP                 =30°+60°                 =90°
Thus, PR is a tangent to circle at R
Distance between center and intersection of tangent i.e., OP = 8cm
 

Page No 118:

Question 7:

Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to ∆ABC with scale factor 32. Justify the construction. Are the two triangles congruent? Note that all the three angles and two sides of the two triangles are equal.
 

Answer:

Step 1: Constant a triangle ABC with AB = 4cm, BC = 6cm and AC = 9cm.
Step 2: Draw a ray AX making an acute angle with the base AC and mark 3 points A1​, A2​, A3​ such that AA1​ = A1​A2 ​= A2​A3
Step 3: Join A2C and draw a line A3​C′ such that A2C is parallel to A3​C′ where C′ lie on the produced AC
Step 4: Now draw another line parallel to BC at C′ such that it meet the produced AB at B′

Here, ΔAB′C′ is the required triangle similar to ΔABC with scale factor 32.




Justification
A3​C′  A2C
ACCC'=21

AC'AC=AC+CC'AC=1+CC'AC=1+12=32
Again BC  B'C'
AB'AB=BC'BC=AC'AC=32
Hence, AB'C'~ABC
 

 



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