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in triangle ABC P, Q AND R ARE the mid points of side AB, BC, CA.prove that
area of parallelogram APQR=( 1/2)* area of triangle[ABC]

  • maths-areas -

    Join PR
    By the mid-point of triangle theorem,
    PR is || to BC and PR = (1/2)BC
    Also triange APR is similar to triange ABC
    so the areas are proportional to the square of their sides.
    since the sides are 1:2
    their areas are 1:4
    so triangle APR = (1/4) of triangle ABC

    Similary BQP would be 1/4 of triangle ABC, and
    RQC is 1/4 of triangle ABC, leaving the inside triangle PQR also as 1/4 of triangle ABC to get 4/4

    so figure APQR = 2/4 or 1/2 of triangle ABC

  • maths-areas -

    How traingle ABC and APR are similar

  • maths-areas -

    Since BC and PR are similar
    angle B = angle APR
    and angle A is common
    if 2 angles of a triangle are equal to 2 corresponding angles of another triangle, the triangles are similar.
    Same argument for the other pairs of similar triangles.

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