**Problem 1: In a ∆ABC, AD is the bisector of ∠A, meeting side BC at D.**

**(i) If BD = 2.5 cm, AB = 5 cm and AC = 4.2 cm, find DC**

**Solution:**

Given:Length of side BD = 2.5 cm, AB = 5 cm, and AC = 4.2 cm.

To find: Length of side DCIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D.

Since, AD is ∠A bisector,

Therefore,

AB/AC = 2.5/DC5/4.2 = 2.5/DC ( Since, AB = 5 cm, and AC = 4.2 cm )

5DC = 4.2 × 2.5

DC = (4.2 × 2.5)/5

DC = 2.1cm

Therefore, Length of side DC is 2.1 cm

**(ii) If BD = 2 cm, AB = 5 cm and DC = 3 cm, find AC**

**Solution:**

Given:Length of side BD = 2 cm, AB = 5 cm, and DC = 3 cm

To find: Length of side ACIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D

Since, AD is ∠A bisector.

Therefore,

AB/AC = BD/DC(since AD is the bisector of ∠A and side BC)5/ AC = 2/3 (Since, BD = 2 cm, AB = 5 cm, and DC = 3 cm )

2AC = 5 × 3

AC = 15/2

AC = 7.5 cm

Therefore ,Length of side AC is 7.5 cm

**(iii) If AB = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD**

**Solution:**

Given:Length of side AB = 3.5 cm, AC = 4.2 cm, and DC = 2.8 cm

To find: Length of side BDIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D

Since, AD is ∠A bisector

Therefore,

⇒ AB/ AC = BD/ DC3.5/ 4.2 = BD/ 2.8 (Since, AB = 3.5 cm, AC = 4.2 cm, and DC = 2.8 cm)

4.2 x BD = 3.5 × 2.8

BD = 7/3

∴ BD = 2.3 cm

Therefore, Length of side BD is 2.3 cm

**(iv) If AB = 10 cm, AC = 14 cm and BC = 6 cm, find BD and DC.**

**Solution:**

Given:Length of side AB = 10 cm, AC = 14 cm, and BC = 6 cm

To find: Length of side BD and DCIn Δ ABC, AD is the bisector of ∠A meeting side BC at D

Since, AD is bisector of ∠A

Therefore,

AB/AC = BD/DC– equation 1Let BD be x, then DC = 6-x

Now, putting values in equation 1

⇒10/ 14 = x/ (6 – x)

14x = 60 – 6x

20x = 60

x = 60/20

∴ BD = 3 cm and DC = (6 – 3) = 3 cm.

Therefore, Length of side BD is 3 cm and DC is 3cm

**(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm, find AB**

**Solution:**

Given:Length of side AC = 4.2 cm, DC = 6 cm, and BC = 10 cm.

To find: Length of side ABIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D.

Since, AD is the bisector of ∠A

Therefore, we get

⇒ AB/ AC = BD/ DCAB/ 4.2 = BD/ 6

We know that,

BD = BC – DC = 10 – 6 = 4 cm

⇒ AB/ 4.2 = 4/ 6

AB = (2 × 4.2)/ 3

∴ AB = 2.8 cm

Therefore, Length of side AB is 2.8 cm

**(vi) If AB = 5.6 cm, AC = 6 cm and DC = 3 cm, find BC**

**Solution:**

Given:Length of side AB = 5.6 cm, BC = 6 cm, and DC = 3 cm

To find: Length of side BCIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D

Since, AD is the ∠A bisector

Therefore, we get

⇒ AB/ AC = BD/ DC5.6/ 6 = BD/ 3

BD = 5.6/ 2 = 2.8cm

And, we know that,

BD = BC – DC

2.8 = BC – 3

2.8 + 3 = BC

∴ BC = 5.8 cm

Therefore, Length of side BC is 5.8 cm

**(vii) If AD = 5.6 cm, BC = 6 cm and BD = 3.2 cm, find AC**

**Solution:**

Given:Length of side AB = 5.6 cm, BC = 6 cm, and BD = 3.2 cm

To find: Length of side ACIn Δ ABC, AD is the bisector of ∠A, meeting side BC at D

Therefore, we get

⇒ AB/ AC = BD/ DC5.6/ AC = 3.2/ DC

And, we know that

BD = BC – DC

3.2 = 6 – DC

∴ DC = 2.8 cm

⇒ 5.6/ AC = 3.2/ 2.8

AC = (5.6 × 2.8)/ 3.2

∴ AC = 4.9 cm

Therefore, Length of side AC is 4.9 cm

**(viii) If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC**

**Solution:**

Given:Length of side AB = 10 cm, AC = 6 cm, and BC = 12 cm

To find: Length of side BD and DCIn Δ ABC, AD is the ∠A bisector, meeting side BC at D.

Since, AD is bisector of ∠A

Therefore, we get

⇒ AB/ AC = BD/ DC10/ 6 = BD/ DC – equation 1

And, we also know that

BD = BC – DC = 12 – DC

Let length of side BD be x,

Then length of side DC will be 12 – x

Now putting values in equation 1, we get

10/ 6 = x/ (12 – x)

5(12 – x) = 3x

60 -5x = 3x

∴ x = 60/8 = 7.5

Hence, DC = 12 – 7.5 = 4.5cm and BD = 7.5 cm

Therefore ,Length of side BD is 7.5 cm and DC is 4.5 cm

**Problem 2: In the figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.**

**Solution:**

Given:Length of side AB = 10 cm, AC = 6 cm and BC = 12 cm

And, AE is the bisector of the exterior ∠CAD

To find: Length of side CESince, AE is the bisector of the exterior ∠CAD

Therefore, we get,

BE / CE = AB / AC ‘ – equation 1

Let length of side CE be x

Therefore, BE = 12+ x

Now, putting this value in equation 1

(12+x)/ x = 10/ 6

6x + 72 = 10x

10x – 6x = 72

4x = 72

∴ x = 18

Since CE = x

Therefore, Length of side CE is 18 cm

**Problem 3 : **Δ**ABC is a triangle such that AB/AC = BD/DC, ∠B = 70**^{o}, ∠C = 50^{o}, find ∠BAD.

^{o}, ∠C = 50

^{o}, find ∠BAD.

**Solution:**

Given:Δ ABC such that AB/AC = BD/DC, ∠B = 70

^{o}and ∠C = 50^{o}

To find: ∠BADIn Δ ABC,

∠A + ∠B + ∠C = 180

^{0}∠A = 180

^{0}– (70^{o}+ 50^{o})= 180

^{o}– 120^{o}= 60

^{o}Since, AB/AC = BD/DC

Therefore, AD is the bisector of ∠A

Therefore, ∠BAD = 1/2 (∠A )

Hence, ∠BAD = 60/2 = 30

^{o}

Therefore, ∠BAD equals to 30^{o}

**Problem 4: Check whether AD is the bisector of ∠A of **Δ**ABC in each of the following:**

**(i) AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm**

**Solution: **

Given:Length of side AB = 5 cm, AC = 10 cm, BD = 1.5 cm and CD = 3.5 cm

To check: whether AD is the bisector of ∠ANow,

AB/AC = 5/10 = 1/2

BD/CD = 1.5/3.5 = 3/7

Therefore,

AB/AC ≠ BD/CD

And since ratio between sides are not proportional

Therefore, AD is not the bisector of ∠A

**(ii) AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm **

**Solution:**

Given:Length of side AB = 4 cm, AC = 6 cm, BD = 1.6 cm and CD = 2.4 cm

To check: whether AD is the bisector of ∠ANow,

AB/AC = 4/6 = 2/3

BD/CD = 1.6/2.4 =2/3

Therefore,

AB/AC = BD/CD

And since ratio between sides are proportional

Therefore, AD is the bisector of ∠A

**(iii) AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm**

**Solution:**

Given:Length of side AB = 8 cm, AC = 24 cm, BD = 6 cm and BC = 24 cm

To check : whether AD is the bisector of ∠ALength of side CD = BC – BD= 24 -6 =18cm

CD = 18cm

Now,

AB/AC = 8/24 = 1/3

BD/CD = 6/18 =1/3

Therefore,

AB/AC = BD/CD

And since ratio between sides are proportional

Therefore, AD is the bisector of ∠A

**(iv) AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm**

**Solution:**

Given:Length of side AB = 6 cm, AC = 8 cm, BD = 1.5 cm and CD = 2 cm

To check: whether AD is the bisector of ∠ANow,

AB/AC = 6/8 = 3/4

BD/CD = 1.5/2 =3/4

Therefore,

AB/AC = BD/CD

And since ratio between sides are proportional

Therefore, AD is the bisector of ∠A

**(v) AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm**

**Solution:**

Given:Length of side AB = 5 cm, AC = 12 cm, BD = 2.5 cm and BC = 9 cm

To check: whether AD is the bisector of ∠ALength of side CD = BC – BD= 9 – 2.5 = 6.5cm

CD = 6.5cm

Now,

AB/AC = 5/12 = 5/12

BD/CD = 2.5/6.5 =5/13

Therefore,

AB/AC ≠ BD/CD

And since ratio between sides are not proportional

Therefore, AD is not the bisector of ∠A

**Problem 5: In fig. AD bisects ∠A, AB = 12 cm, AC = 20 cm, and BD = 5 cm, determine CD.**

**Solution:**

Given:Length of side AB = 12 cm, AC = 20 cm, and BD = 5 cm

AD bisects ∠A

To find: Length of side CDSince, AD is the bisector of ∠A

Therefore, we get

AB/AC = BD/CD

12/20 = 5/CD

12 × CD = 20 × 5

CD = 100/12

CD = 8.33 cm

∴ CD = 8.33 cm.

Therefore, Length of side CD is 8.33 cm

**Problem 6: In **Δ**ABC, if ∠1 = ∠2, **

**Prove that, AB/AC = BD/CD**

**Solution: **

Given:∠1 = ∠2

To prove : AB/AC = BD/CD

Construction: Through C , draw CE || BA which meets BA in E on producing the line further

Proof :Since AD || CE

Therefore, ∠2 = ∠3 (Alternate angle )

And, ∠1 = ∠4 (Corresponding angle)

And ∠1 = ∠2 (Given)

Therefore, ∠3 = ∠4

Since, sides opposite to equal angles are equal

So, AC = AE– equation 1Now, in ΔBCE

AD || CE by construction

So, AD is the bisector of ∠A

Therefore, we get

AB/AE = BD/CD

Since, AC = AE from equation 1

Therefore, AB/AC = BD/CD

Hence proved

**Problem 7 : D and E are the points on sides BC, CA and AB respectively. of a **Δ**ABC such that AD bisects ∠A, BE bisects ∠B and CF bisects ∠C. If AB = 5 cm, BC = 8 cm, and CA = 4 cm, determine AF, CE, and BD.**

**Solution:**

Given:Length of side AB = 5 cm, BC = 8cm, and CA = 4 cm

AD bisects ∠A, BE bisects ∠B and CF bisects ∠C

To find: Length of side AF, CE, and BDSince, AD is the bisector of ∠A

Therefore, we get,

AB/AC = BD/CD

5/4 = BD/ (BC – BD) ( Since CD = BC – BD )

5/4 = BD/ (8 – BD)

40 – 5BD = 4BD

9BD = 40

Therefore, BD = 40/9Since, BE is the bisector of ∠B

Therefore, we get,

AB/BC = AE/EC

5/8 = (AC – EC)/EC ( Since AE = AC – EC )

5/8 = (4 – EC)/EC

5EC = 8(4 – EC)

5EC = 32 -8EC

13EC =32

EC = 32/13

Therefore, EC = 32/13Now, since, CF is the bisector of ∠C

Therefore, we get,

BC/CA = BF/AF

8/4 = (AB – AF)/AF ( Since BF = AB – AF )

2 = (5 – AF)/AF

2AF = 5 – AF

3AF = 5

AF = 5/3

Therefore, AF = 5/3

So length of BD is 40/9cm, EC is 32/13cm and AF is 5/3 cm