Class 10 RD Sharma Solutions - Chapter 4 Triangles - Exercise 4.3
Last Updated :
08 Dec, 2020
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 DC
In Δ ABC, AD is the bisector of ∠A, meeting side BC at D.
Since, AD is ∠A bisector,
Therefore,
AB/AC = 2.5/DC
5/4.2 = 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 AC
In 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 BD
In Δ ABC, AD is the bisector of ∠A, meeting side BC at D
Since, AD is ∠A bisector
Therefore,
⇒ AB/ AC = BD/ DC
3.5/ 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 DC
In 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 1
Let 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 AB
In 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/ DC
AB/ 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 BC
In 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/ DC
5.6/ 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 AC
In Δ ABC, AD is the bisector of ∠A, meeting side BC at D
Therefore, we get
⇒ AB/ AC = BD/ DC
5.6/ 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 DC
In Δ ABC, AD is the ∠A bisector, meeting side BC at D.
Since, AD is bisector of ∠A
Therefore, we get
⇒ AB/ AC = BD/ DC
10/ 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 CE
Since, 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 : ΔABC is a triangle such that AB/AC = BD/DC, ∠B = 70o, ∠C = 50o, find ∠BAD.
Solution:
Given:
Δ ABC such that AB/AC = BD/DC, ∠B = 70o and ∠C = 50o
To find: ∠BAD
In Δ ABC,
∠A + ∠B + ∠C = 1800
∠A = 1800 - (70o + 50o)
= 180o - 120o
= 60o
Since, AB/AC = BD/DC
Therefore, AD is the bisector of ∠A
Therefore, ∠BAD = 1/2 (∠A )
Hence, ∠BAD = 60/2 = 30o
Therefore, ∠BAD equals to 30o
Problem 4: Check whether AD is the bisector of ∠A of ΔABC 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 ∠A
Now,
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 ∠A
Now,
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 ∠A
Length 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 ∠A
Now,
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 ∠A
Length 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 CD
Since, 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 ΔABC, 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 1
Now, 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 ΔABC 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 BD
Since, 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 = 40/9
Since, 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 = 32/13
Now, 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
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