State and prove Basic Proportionality Theorem. Using the above theorem, if ABCD is a trapezium whose diagonals intersect each other at O sho

Question

State and prove Basic Proportionality Theorem. Using the above theorem, if ABCD is a trapezium whose diagonals intersect each other at O show that AO/OC = BO/OD.​

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Remi 4 days 2021-10-14T00:16:50+00:00 2 Answers 0 views 0

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    0
    2021-10-14T00:17:51+00:00

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    Given parameters

    ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.

    To prove

    [tex] \frac{AO}{BO} [/tex] = [tex] \frac{CO}{DO} [/tex]

     

     Construction:

    Draw a line EF passing through O and also parallel to AB

    Now, AB ll CD

    By construction EF ll AB

    ∴ EF ll CD

    Consider the ΔADC,

    Where EO ll AB

    According to basic proportionality theorem

    [tex] \frac{AE}{ED} [/tex] = [tex] \frac{AO}{OC} [/tex]………………………………(1)

    Now consider Δ ABD

    where EO ll AB

    According to basic proportionality theorem

    [tex] \frac{AE}{ED} [/tex] = [tex] \frac{BO}{OD} [/tex]……………………………..(2)

    From equation (1) and (2) we have

    [tex] \frac{AO}{OC} [/tex] = [tex] \frac{BO}{OD} [/tex]

     

     Hence the proof.

    0
    2021-10-14T00:18:17+00:00

    Given parameters

    ABCD is a trapezium where AB || DC and diagonals AC and BD intersect at O.

    To prove

    AO/BO=CO/DO

    Construction:

    Draw a line EF passing through O and also parallel to AB

    Now, AB ll CD

    By construction EF ll AB

    ∴ EF ll CD

    Consider the ΔADC,

    Where EO ll AB

    According to basic proportionality theorem

    Given parameters

    AE/ED=AO/OC………………………………(1)

    Now consider Δ ABD

    where EO ll AB

    According to basic proportionality theorem

    AE/ED=BO/OD……………………………..(2)

    From equation (1) and (2) we have

    AO/OC=BO/OD

    Hence the proof.

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