Question

D, E and F are the points on sides BC, CA and AB respectively of $\triangle ABC$ such that AD bisects $\angle A$, BE bisects $\angle B$and CF bisects $\angle C$. If AB = 5 cm, BC = 8 cm and CA = 4 cm, determine AF, CE and BD.

Answer 1

Answer:

AF = 5/3 cm,  CE = 32/12 cm  and BD = 40/9 cm.

Step-by-step explanation:

Given :  

AB = 5 cm ,  BC = 8 cm ,  CA = 4cm

AD is bisector of ∠A

AB/AC = BD/CD

[The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle]

5/4 = BD/(BC- BD)

5/4 = BD / (8 - BD)

5(8 - BD) = 4 BD

40 - 5 BD = 4BD

40 = 4BD + 5 BD

40 = 9BD

BD = 40/9  cm

BE is bisector of ∠B

AB/BC = AE/EC

[The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle]

⅝ = AC - EC /EC

⅝ = 4 - EC /EC

5EC = 8(4 - EC)

5EC = 32 - 8CE

5CE + 8CE = 32

13CE = 32

CE = 32/13  cm

CF is a bisector of ∠C

BC/CA = BF/CF

[The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle]

8/4 = (AB - AF)/AF

8/4 = (5 - AF)/AF

8AF = 4(5 - AF)

8AF = 20 - 4AF

12AF = 20

AF = 20/12

AF = 5/3 cm

Hence, AF = 5/3 cm,  CE = 32/12 cm  and BD = 40/9 cm.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

it is given that AB=5 BC=8 CA=4

since AD is bisector of A

AB/AC=BD/CD

5÷4=BD/BC-BD

5/4=BD/8-BD

40-5BD=4BD

40=9BD

BD=40/9

BE is bisector of B

AB/BC=AE/EC

5/8=AC-EC/EC

5/8=4-EC/EC

5EC=32-8CE

5CE+8CE=32

13CE=32

CE=32/13

CF is a bisector of C

BC/CA=BF/CF

8/4=AB-AF/AF

8/4=5-AF/AF

8AF=20-4AF

12AF=20

AF=5/3

AF=5/3 CE=32/12 BD=40/9