## In ABC , AD is the bisector of angle A meeting BC at D, CF perpendicular to AB and E is the mid point of AC. What is the median of the trian

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In ABC , AD is the bisector of angle A meeting BC at D, CF perpendicular to AB and E is the mid point of AC. What is the median of the triangle? ​

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1 month 2021-09-21T22:14:06+00:00 1 Answer 0 views 0

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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.

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

(iii) If Ab = 3.5 cm, AC = 4.2 cm and DC = 2.8 cm, find BD.

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

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

(vi) If AB = 5.6 cm, AC = 6 cmand 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.

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Given in △ABC, AD is the bisector of angle A

By internal angle bisector theorem, the bisector of vertical angle of a triangle divides the base in the ratio of the other two sides.

(i)

AC

AB

=

DC

BD

4.2

5

=

DC

2.5

∴ DC=

5

2.5×4.2

∴ DC=2.1cm

(ii)

AC

AB

=

DC

BD

AC

5

=

3

2

∴ AC=

2

5×3

∴ AC=7.5cm

(iii)

AC

AB

=

DC

BD

4.2

3.5

=

2.8

BD

∴ BD=

4.2

3.5×2.8

∴ BD=2.33cm

(iv)

AC

AB

=

DC

BD

Let BD be x then DC becomes 6−x

14

10

=

6−x

x

7

5

=

6−x

x

∴ 30−5x=7x

∴ 12x=30

∴ x=2.5cm

∴ BD=2.5cm and CD=6−2.5=3.5cm

(v)

AC

AB

=

DC

BD

4.2

AB

=

6

10−6

∴ AB=

6

4×4.2

∴ AB=2.8cm

(vi)

AC

AB

=

DC

BD

6

5.6

=

3

BD

∴ BD=

6

5.6×3

∴ BD=2.8cm

⇒ BC=BD+CD=2.8+3=5.8cm

(vii)

AC

AB

=

DC

BD

AC

5.6

=

6−3.2

3.2

∴ AC=

3.2

5.6×2.8

∴ AC=4.9cm

(viii)

AC

AB

=

DC

BD

Let BD be x then DC becomes 6−x

6

10

=

12−x

x

3

5

=

12−x

x

∴ 60−5x=3x

∴ 8x=60

∴ x=7.5cm

⇒ BD=7.5cm and CD=12−7.5=4.5cm