Prove that in a triangle the sum of the medians is less than the perimeter. please answer with a diagram. ​

Question

Prove that in a triangle the sum of the medians is less than the perimeter. please answer with a diagram. ​

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Ivy 3 weeks 2021-11-15T05:00:51+00:00 2 Answers 0 views 0

Answers ( )

  1. Emma
    0
    2021-11-15T05:01:57+00:00

    Answer:

    Figure is in the attachment

    1)

    Let AD,BE & CF be the three medians of a ∆ABC.

    WE KNOW THAT THE SUM OF ANY TWO SIDES OF A TRIANGLE IS GREATER THAN TWICE THE MEDIAN DRAWN TO THE THIRD SIDE.

    AB+AC>2AD;. AB+BC>2BE & BC+AC>2CF.

    Adding We get,

    2(AB+BC+AC) >2(AD+BE+CF)

    (AB+BC+AC) >(AD+BE+CF)

    Hence, the perimeter of a triangle is greater than the sum of its three medians.

    ____________________________

    2)

    Use the result that the perpendicular drawn from a point( outside the line ) to a line is shorter( in length) than a line segment drawn from that point to the line and then add all three cases.

    ___________________________

    Solution:

    Consider ABC is a triangle and AL, BM and CN are the altitudes.

    To Prove:

    AL+BM+CN

    Proof:

    We know that the perpendicular AL drawn from the point A to the line BC is shorter than the line segment AB drawn from the point A to the line BC.

    ALBMCN

    On adding equation 1,2,3

    AL+BM+CN

    ____________________________

    3)

    Given: A ∆ABC in which AD is median.

    To Prove:

    AC+AB>2AD

    Construction:

    Produce AD to E, such that AD=DE.

    Join EC.

    Proof:

    In ∆ADB & ∆ EDC

    AD=ED (by construction)

    angleADB=angleEDC. (vertically opposite angle)

    BD=CD. (D midpoint of BC)

    ∆ADB congruent ∆ EDC (by SAS)

    AB=EC (by CPCT)

    Now ,in ∆AEC, we have AC+EC>AE

    [SINCE, SUM OF ANY TWO SIDES OF A TRIANGLE IS GREATER THAN THE THIRD SIDE]

    AC+EC>AD+DE. (AE=AD+DE)

    AC+AB>2AD (AD=ED & EC=AB)

    Thus, the sum of any two sides of a triangle is greater than twice the median with respect to the third side.

    ================================

    Hope this will help u

    Step-by-step explanation:

    0
    2021-11-15T05:01:59+00:00

    hope this helps you ……………

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