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Show that a median of a triangle divid...

Show that a median of a triangle divides it into two triangles of equal area.

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Solution
Lets assume `ABC` be a triangle and Let `AD` be one of its medians.
In `/_\ ABD` and `/_\ ADC` the vertex is common and bases are equal.
Constructing `AE_|_ BC`.
Now area `(/_\ABD)`
`=1/2` `xx` base `xx` altitude of `/_\ ADB`
`=1/2 xx BD xx AE=1/2 xx DC xx AE(:'BD=DC)`
but `DC` and `AE` are the base and altitude of `/_\ ACD`
`= 1/2` `xx` base `DC` `xx` altitude of `/_\ ACD`
`=area/_\ ACD`
`=>area(/_\ ABD)=area(/_\ ACD)`
Hence proved.
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Knowledge Check

  • The median of a triangle divides it into two

    A
    triangles of equal area
    B
    congruent triangles
    C
    right angled triangles
    D
    isosceles triangles

  • Assertion (A) : If ABCD is a rhombus whose one angle is 60^(@) then the ratio of the lengths of its diagonals is sqrt3 : 1 Reason (R ) : Median of a triangle divides it into two triangle of equal area.

    A
    Both Assertion (A) and Reason (R ) are true and Reason (R ) is a correct explansion of Assertion (A).
    B
    Both Assertion (A) and Reason (R ) are true but Reason (R ) is not a correct explansion of Assertion (A).
    C
    Assertion (A) is true and Reason (R ) is false.
    D
    Assertion (A) is false and Reason (R ) is true.

  • Consider the following statements in respect of any triangle I. The three medians of a triangle divide it into six triangles of equal area. II. The perimeter of a triangle is greater than the sum of the lengths of its three medians. Which of the statements given above is/are correct ?

    A
    I only
    B
    II only
    C
    Both I and II
    D
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