A line segment AB of length 10 cm is divided at P internally in the ratio 2:3. Then find the length of AP ​

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A line segment AB of length 10 cm is divided at P internally in the ratio 2:3. Then find the length of AP

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Mary 4 weeks 2021-08-18T19:41:26+00:00 1 Answer 0 views 0

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    2021-08-18T19:43:16+00:00

    Step-by-step explanation:

    We follow the following step of construction.

    Steps of construction

    Step I

    Drawn a line segment AB=10 cm by using a ruler.

    Step II

    Drawn any ray making an acute angle ∠BAX with AB.

    Step III

    Along AX, mark-off 5(=3+2) points A

    1

    ,A

    2

    ,A

    3

    ,A

    4

    and A

    5

    such that

    AA

    1

    =A

    1

    A

    2

    =A

    3

    A

    4

    =A

    4

    A

    5

    .

    Step Iv

    Join BA

    5

    Step v

    Through A

    3

    draw a line A

    3

    P parallel to A

    5

    B by making an angle equal to ∠AA

    5

    B at A

    3

    intersecting AB at a point P.

    The point P so obtained is the required point, which divides AB internally in the ration 3:2.

    ALTERNATIVE METHOD FOR DIVISION OF A LINE SEGMENT INTERNALLY IN A GIVEN RATIO

    We may use the following steps to divide a given line segment AB internally in a given ratio m:n, where m and n are natural numbers.

    Steps of construction

    Step I

    Draw line segment AB of given length.

    Step II

    Draw any ray AX making an acute angle ∠BAX with AB.

    Step III

    Draw a ray BY, on opposite side of AX, parallel to AX by making an angle ∠BAY equal to ∠BAX.

    Step IV

    Mark off m points A

    1

    ,A

    2

    ,,A

    m

    , on AX and n points B

    1

    ,B

    2

    ,,B

    n

    on BY such that

    AA

    1

    =A

    1

    A

    2

    =.=A

    m−1

    A

    m

    =BB

    1

    =B

    1

    B

    2

    =.=Bn−1B

    n

    .

    Step V

    Join A

    m

    B

    n

    . Suppose it intersects AB at P.

    The point P is the required point dividing AB in the ratio m:n.

    In triangles AA

    m

    P and BB

    n

    P, we have

    ∠A

    m

    AP=∠PBB

    n

    and, APA

    m

    =∠BPB

    n

    So, by AA similarly criterion, we have

    △A A

    m

    P−△BB

    n

    P

    BB

    n

    AA

    m

    =

    BP

    AP

    BP

    AP

    =

    n

    m

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