If a ∆ ABC, AD is the bisector of ∠A, meeting side BC at D.
(v) If AC = 4.2 cm, DC = 6 cm and BC = 10 cm. find AB.
(vi) If AB = 5.6 cm, AC = 6 cm and DC = 3 cm, find BC.
(vii) If AD = 5.6 cm, BC = 6 cm and BD = 3.2 cm, find AC.
(viii) If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC.
Answers
SOLUTION :
(v) Given : AC = 4.2 cm, DC = 6 cm, and BC = 10 cm.
In Δ ABC, AD is the bisector of ∠ A, meeting side BC at D.
Since, AD is the bisector of ∠ A
Therefore, AC/AB = DC/BD
[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
Then, 4.2/AB = 6/4
6 AB = 4.2 x 4
AB = (4.2 x 4)/6
AB =0 .7 × 4
AB = 2.8 cm
Hence, AB = 2.8 cm
(vi) Given : AB = 5.6 cm, AC = 6 cm, and DC = 3 cm
In Δ ABC, AD is the bisector of ∠ A, meeting side BC at D
Since, AD is the ∠ A bisector
Therefore, AB/AC = BD/DC
[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
Then, 5.6/6 = BD/3
5.6× 3 = 6×BD
BD = (5.6 × 3)/ 6
BD = 5.6/2
BD = 2.8 cm
BC = BD + DC
BC = 2.8 + 3
BC = 5.8 cm
Hence, BC = 5.8 cm
(vii) Given : AB = 5.6 cm, BC = 6 cm, and BD = 3.2 cm
DC = BC – BD
DC = 6 - 3.2
DC = 2.8
In Δ ABC, AD is the bisector of ∠ A , meeting side BC at D.
Therefore, AB/AC = BD/DC
[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
5.6/AC=3.2/2.8
AC = (5.6 x 2.8)/3.2
AC = 7 × .7
AC = 4.9 cm
Hence, AC = 4.9 cm
(viii) Given : AB = 10 cm, AC = 6 cm, and BC = 12 cm
Let BD = x cm
DC = BC - BD
DC = (12 - x) cm
In Δ ABC, AD is the ∠ A bisector, meeting side BC at D.
Since, AD is bisector of ∠ A
So, AC/AB = DC/BD
[The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle]
Then,
6/10 = (12–x)/x
6x = 10(12- x)
6x = 120 - 10x
6x +10x = 120
16x = 120
x = 120/16
x = 7.5
BD = 7.5 cm
DC = BC - BD
DC = 12 - BD
DC = 12 - 7.5
DC = 4.5
Hence,BD = 7.5 cm and DC = 4.5 cm.
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Answer:
(vii) Question should be "In a ΔABC, AD is the bisector of angleA, meeting side BC at D. if AB=5.6 cm, BC=6 cm, BD=3.2 cm, find AC?
Solution
ΔABC, AD is the bisector of angle A AB=5.6 cm, BC=6 cm, BD=3.2 cm Hence DC = BC – BD = 6 – 3.2 = 2.8 cm By internal angle bisector theorem, the bisector of vertical angle of a triangle divides the base in the ratio of the other two sides”