Chapter 9 : Congruence of Triangles and Inequalities in a Triangle - R S Aggarwal And V Aggarwal Solutions for Class 9 Maths CBSE

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Chapter 9 - Congruence of Triangles and Inequalities in a Triangle Excercise MCQ

Question 1

Which of the following is not a criterion for congruence of triangles?

(a) SSA

(b) SAS

(c) ASA

(d)SSS

Solution 1

Correct option: (a)

SSA is not a criterion for congruence of triangles.

Question 2

If AB = QR, BC = RP and CA = PQ, then which of the following holds?

(a) ABC PQR

(b) CBA  PQR

(c) CAB  PQR

(d) BCA PQR

Solution 2

Correct option: (c)

  

Question 3

If ABC PQR then which of the following is not true?

(a) BC = PQ

(b) AC = PR

(c) BC = QR

(d) AB = PQ

Solution 3

  

Question 4

In Δ ABC, AB = AC and B = 50°. Then, A = ?

(a) 40° 

(b) 50° 

(c) 80° 

(d) 130° 

Solution 4

Correct option: (c)

In ΔABC,

AB = AC

C = B (angles opposite to equal sides are equal)

C = 50° 

Now, A + B + C = 180° 

A + 50° + 50° = 180° 

A = 80° 

Question 5

In Δ ABC, BC = AB and B = 80°. Then, A = ?

(a) 50° 

(b) 40° 

(c) 100° 

(d) 80° 

Solution 5

Correct option: (a)

In ΔABC,

BC = AB

A = C (angles opposite to equal sides are equal)

Now, A + B + C = 180° 

A + 80° + A = 180° 

2A = 100° 

A = 50°  

Question 6

In ΔABC, C = A, BC = 4 cm and AC = 5 cm. Then, AB = ?

(a) 4 cm

(b) 5 cm

(c) 8 cm

(d) 2.5 cm

Solution 6

Correct option: (a)

In ΔABC,

C = A

AB = BC (sides opposite to equal angles are equal)

AB = 4 cm 

Question 7

Two sides of a triangle are of length 4 cm and 2.5 cm. The length of the third side of the triangle cannot be

(a) 6 cm

(b) 6.5 cm

(c) 5.5 cm

(d) 6.3 cm

Solution 7

Correct option: (b)

The sum of any two sides of a triangle is greater than the third side.

Since, 4 cm + 2.5 cm = 6.5 cm

The length of third side of a triangle cannot be 6.5 cm. 

Question 8

In ΔABC, if C > B, then

(a) BC > AC

(b) AB > AC

(c) AB < AC

(d) BC < AC

Solution 8

Correct option: (b)

We know that in a triangle, the greater angle has the longer side opposite to it.

In ΔABC,

C > B

AB >AC  

Question 9

It is given that ABC FDE in which AB = 5 cm, B = 40o, A = 80o and FD = 5 cm. Then which of the following is true?

(a) D = 60o

(b) E = 60o

(c) F = 60o

(d) D = 80o

Solution 9

  

  

Question 10

In ABC, A = 40o and B = 60o. Then the longest side of ABC is

(a) BC

(b) AC

(c) AB

(d) Cannot be determined

Solution 10

  

  

Question 11

In the given figure AB > AC. Then, which of the following is true?

(a) AB < AD

(b) AB = AD

(c) AB > AD

(d) Cannot be determined

 

  

Solution 11

Correct option: (c)

 



Question 12

In the given figure AB > AC. If BO and CO are the bisectors of B and C respectively, then

(a) OB = OC

(b) OB > OC

(c) OB < OC

 

 

Solution 12

  

Question 13

In the given figure, AB = AC and OB = OC. Then, ABO : ACO = ?

(a) 1 :1

(b) 2 : 1

(c) 1 :2

(d) None of these 

  

Solution 13

  

Question 14

If the altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is

(a) Equilateral

(b) Isosceles

(c) Scalene

(d) Right-angled

Solution 14

  

Question 15

In ABC and DEF, it is given that AB = DE and BC = EF. In order that ABC DEF, we must have

(a) A = D

(b) B = E

(c) C = F

(d) None of these

  

Solution 15

  

Question 16

In ABC and DEF, it is given that B = E and C = F. In order that ABC DEF, we must have

(a) AB = DF

(b) AC = DE

(c) BC = EF

(d) A = D

  

Solution 16

  

Question 17

In ABC and PQR, it is given that AB = AC, C = P and B = Q. Then, the two triangles are

(a) Isosceles but not congruent

(b) Isosceles but congruent

(c) Congruent but not isosceles

(d) Neither congruent nor isosceles

  

Solution 17

  

Question 18

Which is true ?

(a) A triangle can have two right angles.

(b) A triangle can have two obtuse angles.

(c) A triangle can have two acute angles.

(d) An exterior angle of a triangle is less than either of the interior opposite angles.

  

Solution 18

  

Question 19

Fill in the blanks with

(a) (Sum of any two sides of a triangle)……(the third side)

(b) (Difference of any two sides of a triangle)…..(the third side)

(c) (Sum of three altitudes of a triangle) ……. (sum of its three sides

(d) (Sum of any two sides of a triangle)….. (twice the median to the 3rd side)

(e) (Perimeter of a triangle)……(sum of its medians)

Solution 19

  

Question 20

Fill in the blanks

(a) Each angle of an equilateral triangles measures …….

(b) Medians of an equilateral triangle are ……….

(c) In a right triangle the hypotenuse is the ….. side

(d) Drawing a ABC with AB = 3cm, BC= 4 cm and CA = 7 cm is ……..

Solution 20

  

Chapter 9 - Congruence of Triangles and Inequalities in a Triangle Excercise Ex. 9A

Question 1

In the given figure, AB CD and O is the midpoint of AD.

Show that (i) Δ AOB Δ DOC (ii) O is the midpoint of BC.

 

  

Solution 1

(i) In ΔAOB and ΔDOC,

BAO = CDO (AB CD, alternate angles)

AO = DO (O is the mid-point of AD)

AOB = DOC (vertically opposite angles)

ΔAOB ΔDOC (by ASA congruence criterion)

(ii) Since ΔAOB ΔDOC,

BO = CO (c.p.c.t.)

O is the mid-point of BC.

Question 2

In the given figure, AD and BC are equal perpendiculars to a line segment AB. Show that CD bisect AB.

 

  

Solution 2

In ΔAOD and ΔBOC,

AOD = BOC (vertically opposite angles)

DAO = CBO (Each 90°)

AD = BC (given)

ΔAOD BOC (by AAS congruence criterion)

AO = BO (c.p.c.t.)

CD bisects AB. 

Question 3

In the given figure, two parallels lines l and m are intersected by two parallels lines p and q. Show that Δ ABC ≅ Δ CDA.

 

  

Solution 3

In ΔABC and ΔCDA

BAC = DCA  (alternate interior angles for p q)

AC = CA  (common)

BCA = DAC (alternate interior angles for l m)

ΔABC ΔCDA (by ASA congruence rule)

Question 4

AD is an altitude of an isosceles ΔABC in which AB = AC.

Show that (i) AD bisects BC, (ii) AD bisects A.

 

  

Solution 4

(i) In ΔBAD and ΔCAD

ADB = ADC  (Each 90° as AD is an altitude)

AB = AC (given)

AD = AD (common)

 ΔBAD ΔCAD (by RHS Congruence criterion)

BD = CD (c.p.c.t.)

Hence AD bisects BC.

 

(ii) Also, BAD = CAD (c.p.c.t.)

 Hence, AD bisects A.

Question 5

In the given figure, BE and CF are two equal altitudes of ΔABC.

Show that (i) ΔABE ΔACF, (ii) AB = AC.

 

  

Solution 5

(i) In ΔABE and ΔACF,

AEB = AFC (Each 90°)

BE = CF (given)

BAE = CAF (common A)

ΔABE ACF (by ASA congruence criterion)

(ii) Since ΔABE ≅ ΔACF,

AB = AC (c.p.c.t.)

Question 6

ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC. If AD is extended to intersect BC at E, show that

(i) ΔABD ΔACD

(ii) ΔABE ΔACE

(iii) AE bisects A as well as D

(iv) AE is the perpendicular bisector of BC.

 

  

Solution 6

(i) In ΔABD and ΔACD,

AB = AC  (equal sides of isosceles ΔABC)

DB = DC  (equal sides of isosceles ΔDBC)

AD = AD (common)

ΔABD ACD (by SSS congruence criterion)

 

(ii) Since ΔABD ≅ ΔACD,

BAD = CAD (c.p.c.t.)

BAE = CAE ….(1)

Now, in ΔABE and ΔACE

AB = AC (equal sides of isosceles ΔABC)

BAE = CAE [From (1)]

AE = AE (common) 

ΔABE ACE (by SAS congruence criterion)

 

(iii) Since ΔABD ≅ ΔACD,

BAD = CAD (c.p.c.t.)

BAE = CAE 

Thus, AE bisects A.

In ΔBDE and ΔCDE,

BD = CD (equal sides of isosceles ΔABC)

BE = CE (c.p.c.t. since ΔABE ACE) 

DE = DE (common)

ΔBDE CDE (by SSS congruence criterion)

BDE = CDE (c.p.c.t.)

Thus, DE bisects D, i.e., AE bisects D.

Hence, AE bisects A as well as D.

 

(iv) Since ΔBDE ≅ ΔCDE,

BE = CE and BED = CED (c.p.c.t.)

BE = CE and BED = CED = 90° (since BED and CED form a linear pair) 

DE is the perpendicular bisector of BC.

AE is the perpendicular bisector of BC.

Question 7

In the given figure, if x = y and AB = CB, then prove that AE = CD.

Solution 7

 

 


Question 8

In the given figure, line l is the bisector of an angle A and B is any point on l. If BP and BQ are perpendiculars from B to the arms of A, show that

(i) ΔAPB ΔAQB

(ii) BP = BQ, i.e., B is equidistant from the arms of A.

 

  

Solution 8

(i) In ΔAPB and ΔAQB,

APB = AQC (Each 90°)

BAP = BAQ (line l is the bisector of A)

AB = AB (common)

ΔAPB AQB (by AAS congruence criterion)

(ii) Since ΔAPB ≅ ΔAQB,

BP = BQ (c.p.c.t.) 

Question 9

ABCD is a quadrilateral such that diagonal AC bisect the angles A and C. Prove that AB = AD and CB = CD.

Solution 9

  

In ΔABC and ΔADC,

BAC = DAC (AC bisects A)

AC = AC (common)

BCA = DCA (AC bisects C)

ΔABC ADC (by ASA congruence criterion)

AB = AD and CB = CD (c.p.c.t.) 

Question 10

ΔABC is a right triangle right angled at A such that AB = AC and bisector of C intersect the side AB at D. Prove that AC + AD = BC.

Solution 10

Construction: Draw DE BC.

  

In ΔDAC and ΔDEC,

DAC = DEC (Each 90°)

DCA = DCE (CD bisects C)

CD = CD (common)

ΔDAC ≅ ΔDEC (by AAS congruence criterion)

DA = DE (c.p.c.t.) ….(i)

and AC = EC (c.p.c.t.)  ….(ii)

 

Given, AB = AC

B = C (angles opposite to equal sides are equal)

 

In ΔABC, by angle sum property,

A + B + C = 180° 

90° + B + B = 180° 

2B = 90° 

B = 45° 

 

In ΔBED,

BDE + B = 90° (since BED = 90°)

BDE + 45° = 90° 

BDE = 45° 

BDE = DBE = 45° 

DE = BE ….(iii)

From (i) and (iii),

DA = DE = BE ….(iv) 

 

Now, BC = BE + EC

BC = DA + AC [From (ii) and (iv)

AC + AD = BC

Question 11

In the given figure, OA = OB and OP = OQ. Prove that (i) PX = QX (ii) AX = BX.

Solution 11

Question 12

In the given figure, ABC is an equilateral triangle; PQ || AC and AC is produced to R such that CR = BP. Prove that QR bisects PC.

Solution 12


 


Question 13

In the given figure, ABCD is a quadrilateral in which AB || DC and P is the midpoint BC. On producing, AP and DC meet at Q. Prove that (i) AB = CQ, (ii) DQ = DC + AB.

Solution 13


Question 14

In the given figure, ABCD is square and P is a point inside it such that PB = PD. Prove that CPA is a straight line.

Solution 14

 


Question 15

In the given figure, O is a point in the interior of square ABCD such that ΔOAB is an equilateral triangle. Show that ΔOCD is an isosceles triangle.

 

  

Solution 15

ΔOAB is an equilateral triangle.

OAB = OBA = AOB = 60° 

 

ABCD is a square.

A = B = C = D = 90° 

 

Now, A = DAO + OAB

90° = DAO + 60° 

DAO = 90° - 60° = 30° 

 

Similarly, CBO = 30° 

 

In ΔOAD and ΔOBC,

AD = BC  (sides of a square ABCD)

DAO = CBO = 30° 

OA = OB (sides of an equilateral ΔOAB)

ΔOAD ≅ ΔOBC (by SAS congruence criterion)

OD = OC (c.p.c.t.)

Hence, ΔOCD is an isosceles triangle.

Question 16

In the adjoining figure, X and Y are respectively two points on equal sides AB and AC of ABC such that AX = AY. Prove that CX = BY.

Solution 16


Question 17

In ABC, D is the midpoint of BC. If DL AB and DM AC such that DL = DM, prove that AB = AC.

Solution 17

 


Question 18

In ABC, AB = AC and the bisectors of B and C meet at a point O. Prove that BO = CO and the ray AO is the bisector A.

Solution 18


 


Question 19

The line segments joining the midpoints M and N of parallel sides AB and DC respectively of a trapezium ABCD is perpendicular to both the sides AB and DC. Prove that AD = BC.

Solution 19

Construction: Join AN and BN.

  

In ΔANM and ΔBNM

AM = BM (M is the mid-point of AB)

AMN = BMN (Each 90°)

MN = MN (common)

ΔANM ΔBNM (by SAS congruence criterion)

AN = BN (c.p.c.t.) ….(i)

And, ANM = BNM (c.p.c.t.)

90° - ANM = 90° - BNM

AND = BNC ….(ii) 

 

In ΔAND and DBNC,

AN = BN [From (i)]

AND = BNC [From (ii)]

DN = CN (N is the mid-point of DC)

ΔAND ΔBNC (by SAS congruence criterion)

AD = BC (c.p.c.t.)

Question 20

The bisectors of B and C of an isosceles triangle with AB = AC intersect each other at a point O. BO is produced to meet AC at a point M. Prove that MOC = ABC.

Solution 20

  

In ΔABC, AB = AC

ABC = ACB

OBC = OCB ….(i)

 

Now, by exterior angle property,

MOC = OBC + OCB

MOC = 2OBC [From (i)]

MOC = ABC (OB is the bisector of ABC)  

Question 21

The bisectors of B and C of an isosceles ΔABC with AB = AC intersect each other at a point O. Show that the exterior angle adjacent to ABC is equal to BOC.

Solution 21

  

In ΔABC, AB = AC

ABC = ACB

OBC = OCB ….(i)

In ΔBOC, by angle sum property,

BOC + OBC + OCB = 180° 

BOC + 2OBC = 180° [From (i)]

BOC + ABC = 180° 

BOC + (180° - ABP) = 180° (ABC and ABP form a linear pair)

BOC + 180° - ABP = 180° 

BOC - ABP = 0

BOC = ABP 

Question 22

P is a point on the bisector of ABC. If the line through P, parallel to BA meets BC at Q, prove that ΔBPQ is an isosceles triangle.

Solution 22

  

AB PQ and BP is a transversal.

ABP = BPQ (alternate angles) ….(i)

BP is the bisector of ABC.

ABP = PBC

ABP = PBQ ….(ii)

From (i) and (ii), we have

BPQ = PBQ

PQ = BQ (sides opposite to equal angles are equal)

ΔBPQ is an isosceles triangle.

Question 23

The image of an object placed at a point A before a plane mirror LM is seen at the point B by an observer at D, as shown in the figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.

 

  

Solution 23

To prove that the image is as far behind the mirror as the object is in front of the mirror, we need to prove that AT = BT. 

 

We know that angle of incidence = angle of reflection.

ACN = DCN ….(i)

 

AB CN and AC is the transversal.

TAC = ACN (alternate angles) ….(ii)

Also, AB CN and BD is the transversal.

 

TBC = DCN (corresponding angles) ….(iii)

 

From (i), (ii) and (iii),

TAC = TBC ….(iv)

 

In ΔACT and ΔBCT,

TAC = TBC [From (iv)]

ATC = BTC (Each 90°)

CT = CT (common)

ΔACT ΔBCT (by AAS congruence criterion)

AT = BT (c.p.c.t.)

Question 24

In the adjoining figure, explain how one can find the breadth of the river without crossing it.

Solution 24

Let AB be the breadth of a river. Now take a point M on that bank of the river where point B is situated. Through M draw a perpendicular and take point N on it such that point, A, O and N lie on a straight line where point O is the mid point of BM.

Question 25

In a ΔABC, D is the midpoint of side AC such that BD =  . Show that ABC is a right angle.

Solution 25

  

D is the mid-point of AC.

AD = CD =   

Given, BD =   

AD = CD = BD 

 

Consider AD = BD

BAD = ABD (i)(angles opposite to equal sides are equal)

Consider CD = BD

BCD = CBD (ii)(angles opposite to equal sides are equal)

 

In ΔABC, by angle sum property,

ABC + BAC + BCA = 180° 

ABC + BAD + BCD = 180° 

ABC + ABD + CBD = 180° [From (i) and (ii)]

ABC + ABC = 180° 

2ABC = 180° 

ABC = 90° 

Hence, ABC is a right angle.

Question 26

"If two sides and an angle of one triangle are equal to two sides and an angle of another triangle then the two triangles must be congruent." Is the statement true? Why?

Solution 26

The given statement is not true.

Two triangles are congruent if two sides and the included angle of one triangle are equal to corresponding two sides and the included angle of another triangle. 

Question 27

"If two angles and a side of one triangle are equal to two angles and a side of another triangle then the two triangles must be congruent." Is the statement true? Why?

Solution 27

The given statement is not true.

Two triangles are congruent if two angles and the included side of one triangle are equal to corresponding two angles and the included angle of another triangle. 

Chapter 9 - Congruence of Triangles and Inequalities in a Triangle Excercise Ex. 9B

Question 1

Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.

5 cm, 4 cm, 9 cm

Solution 1

No, it is not possible to construct a triangle with lengths of its sides given because the sum of two sides, 5 cm and 4 cm, is not greater than the third side, 9 cm. 

Question 2

Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.

8 cm, 7 cm, 4 cm

Solution 2

Yes, it is not possible to construct a triangle with lengths of its sides given because the sum of any two sides is greater than the third side. 

Question 3

Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.

10 cm, 5 cm, 6 cm

Solution 3

Yes, it is not possible to construct a triangle with lengths of its sides given because the sum of any two sides is greater than the third side. 

Question 4

Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.

2.5 cm, 5 cm, 7 cm

Solution 4

Yes, it is not possible to construct a triangle with lengths of its sides given because the sum of any two sides is greater than the third side. 

Question 5

Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.

3 cm, 4 cm, 8 cm

Solution 5

No, it is not possible to construct a triangle with lengths of its sides given because the sum of two sides, 3 cm and 4 cm, is not greater than the third side, 8 cm. 

Question 6

In ΔABC, A = 50° and B = 60°. Determine the longest and shortest sides of the triangle.

Solution 6

In ΔABC,

A + B + C = 180° 

50° + 60° + C = 180° 

C = 70° 

Thus, we have

A < B < C

BC < AC < AB

Hence, the longest side is AB and the shortest side is BC. 

Question 7

In ΔABC, A = 100° and C = 50°. Which is its shortest side?

Solution 7

In ΔABC,

A + B + C = 180° 

100° + B + 50° = 180° 

B = 30° 

Thus, we have

B < C < A

AC < AB < BC

Hence, the shortest side is AC. 

Question 8

In ABC, if A = 90o, which is the longest side?

Solution 8

Question 9

In ABC, if A = B = 45o, name the longest side.

Solution 9

Question 10

In ABC, side AB is produced to D such that BD = BC. If B = 60o and A = 70o, prove that (i) AD > CD and (ii) AD > AC.

Solution 10

Question 11

In the given figure, B < A and C < D. Show that AD < BC.

 

  

Solution 11

In ΔAOB,

B < A

AO < BO ….(i)

In ΔCOD,

C < D

DO < CO ….(ii)

Adding (i) and (ii),

AO + DO < BO + CO

AD < BC 

Question 12

AB and CD are respectively the smallest and largest sides of quadrilateral ABCD. Show that A > C and B > D.

 

  

Solution 12

Construction: Join AC and BD.

  

In ΔABC,

BC > AB

BAC > ACB ….(i)

In ΔACD,

CD > AD

CAD > ACD ….(ii)

Adding (i) and (ii), we get

BAC + CAD > ACB + ACD

A > C 

 

In ΔADB,

AD > AB

ABD > ADB ….(iii)

In ΔBDC,

CD > BC

CBD > BDC ….(iv)

Adding (iii) and (iv), we get

ABD + CBD > ADB + BDC

B > D 

Question 13

In a quadrilateral ABCD, show that

(AB + BC + CD + DA) > (AC + BD).

Solution 13

  

In ΔABC,

AB + BC > AC  ….(i)

In ΔACD,

DA + CD > AC  ….(ii)

In ΔADB,

DA + AB > BD  ….(iii)

In ΔBDC,

BC + CD > BD  ….(iv)

Adding (i), (ii), (iii) and (iv), we get

AB + BC + DA + CD + DA + AB + BC + CD > AC + AC + BD + BD

2(AB + BC + CD + DA) > 2(AC + BD)

AB + BC + CD + DA > AC + BD 

Question 14

In a quadrilateral ABCD, show that

(AB + BC + CD + DA) < 2(BD + AC).

Solution 14

  

In ΔAOB,

AO + BO > AB  ….(i)

In ΔBOC,

BO + CO > BC  ….(ii)

In ΔCOD,

CO + DO > CD  ….(iii)

In ΔAOD,

DO + AO > DA  ….(iv)

Adding (i), (ii), (iii) and (iv), we get

AO + BO + BO + CO + CO + DO + DO + AO > AB + BC + CD + DA

2(AO + CO) + 2(BO + DO) > AB + BC + CD + DA

2AC + 2BD > AB + BC + CD + DA

2(AC + BD) > AB + BC + CD + DA

AB + BC + CD + DA < 2(AC + BD) 

Question 15

In ABC, B = 35o, C = 65o and the bisector of BAC meets BC in X. Arrange AX, BX and CX in descending order.

 

Solution 15

 


 

Question 16

In the given figure, PQ > PR and QS and RS are the bisectors of Q and R respectively. Show that SQ > SR.

 

  

Solution 16

In ΔPQR,

PQ > PR

PRQ > PQR

SRQ > SQR

SQ > SR

Question 17

D is any point on the side AC of ΔABC with AB = AC. Show that CD < BD.

Solution 17

  

In ΔABC,

AB = AC

ABC = ACB ….(i)

Now, ABC = ABD + DBC

ABC > DBC

ACB > DBC [From (i)]

DCB > DBC

BD > CD

i.e. CD < BD 

Question 18

Prove that in a triangle, other than an equilateral triangle, angle opposite to the longest side is greater than   of a right angle.

Solution 18

  

Let PQR be the required triangle.

Let PR be the longest side.

Then, PR > PQ

Q > R ….(i)

Also, PR > QR

Q > P ….(ii)

Adding (i) and (ii), we get

2Q > R + P

2Q + Q > P + Q + R (adding Q to both sides)

3Q > 180° 

Q > 60° 

Question 19

In the given figure, prove that CD + DA + AB > BC

  

Solution 19

In ΔCDA,

CD + DA > AC ….(i)

In ΔABC,

AC + AB > BC ….(ii)

Adding (i) and (ii), we get

CD + DA + AC + AB > AC + BC

Subtracting AC from both sides, we get

CD + DA + AB > BC 

Question 20

In the given figure, prove that

CD + DA + AB + BC > 2AC.

  

Solution 20

In ΔCDA,

CD + DA > AC ….(i)

In ΔABC,

AB + BC > AC ….(ii)

Adding (i) and (ii), we get

CD + DA + AB + BC > AC + AC

CD + DA + AB + BC > 2AC 

Question 21

If O is a point within ABC, show that:

AB + AC > OB + OC

Solution 21

Given : ABC is a triangle and O is appoint insideit.

 

To Prove : (i) AB+AC > OB +OC

Question 22

If O is a point within ABC, show that:

AB + BC + CA > OA + OB + OC

Solution 22

AB+BC+CA > OA+OB+OC

Question 23

If O is a point within ABC, show that:

OA + OB + OC > (AB + BC + CA)

Solution 23

OA+OB+OC> (AB+BC+CA)

Proof:

(i)InABC,

AB+AC>BC.(i)

And in , OBC,

OB+OC>BC.(ii)

Subtracting (i) from (i) we get

(AB+AC)-(OB+OC)> (BC-BC)

i.e.AB+AC>OB+OC

 

(ii)AB+AC> OB+OC[proved in (i)]

Similarly,AB+BC > OA+OC

AndAC+BC> OA +OB

Addingboth sides of these three inequalities, we get

(AB+AC) +(AC+BC) +(AB+BC)>OB+OC+OA+OB+OA+OC

i.e.2(AB+BC+AC)> 2(OA+OB+OC)

Therefore, we have

AB+BC+AC > OA+OB+OC

 

(iii)InOAB

OA+OB > AB(i)

InOBC,

OB+OC > BC(ii)

And, in OCA,

OC+OA>CA

Adding (i), (ii) and (iii)we get

(OA+OB) + (OB+OC)+(OC+OA)> AB+BC+CA

i.e2(OA+OB+OC) > AB+BC+CA

OA+OB+OC> (AB+BC+CA)

Question 24

In the given figure, AD BC and CD > BD. Show that AC > AB.

 

  

Solution 24

Construction: Mark a point S on BC such that BD = SD. Join AS.

  

In ΔADB and ΔADS,

BD = SD (by construction)

ADB = ADS   (Each equal to 90°)

AD = AD (common)

ΔADB ΔADS (by SAS congruence criterion)

AB = AS (c.p.c.t.)

Now, in ΔABS,

AB = AS

ASB = ABS ….(i)(angles opposite to equal sides are equal)

In ΔACS,

ASB > ACS ….(ii)

From (i) and (ii), we have

ABS > ACS

ABC > ACB

AC > AB

Question 25

In the given figure, D is a point on side BC of a ΔABC and E is a point such that CD = DE. Prove that AB + AC > BE.

 

  

Solution 25

In ΔABC,

AB + AC > BC

AB + AC >BD + DC

AB + AC >BD + DE ….(i) [since CD = DE]

In ΔBED,

BD + DE > BE ….(ii)

From (i) and (ii), we have

AB + AC > BE

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