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Trapezium diagonals divide each other in the ratio of the parallel sides

This type of question, asked 3 times: CDS 1 2017, CDS 2 2020, CDS 1 2021. All 3 questions below, oldest first.

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  1. Question 1

    CDS 1 2017

    86
    In a trapezium ABCD, AB is parallel to CD and the diagonals intersect each other at O. What is the ratio of OA to OC equal to ?

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  2. Question 2

    CDS 2 2020

    65
    Consider the following statements : 1. The diagonals of a trapezium divide each other proportionally. 2. Any line drawn parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally. Which of the above statements is/are correct ?

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  3. Question 3

    CDS 1 2021

    76
    In a trapezium ABCDABCD, ABAB is parallel to DCDC. The diagonals ACAC and BDBD intersect at PP. If AP:PC=4:(4x−4)AP : PC = 4 : (4x - 4) and BP:PD=(2x−1):(2x+4)BP : PD = (2x - 1) : (2x + 4), then what is the value of xx?

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