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

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    2021-09-21T22:15:10+00:00

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    MATHS

    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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    ANSWER

    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

    [ AB=AD ]

    ∴ 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

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