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If b is the mean proportional between a and c, prove that a, c, a2 + b2 and b2 + c2 are proportional.

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This is an important ques from the Book ML Aggarwal class 10th,, chapter – 7, ratio and proportion.

It is given that b is the mean proportional between a and c, and we have to  prove that a, c, a2 + b2 and b2 + c2 are proportional.

Question 12, exercise 7.2

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  1. Solution:

    It is given that

    b is the mean proportional between a and c

    We can write it as

    b2 = a × c

    b2 = ac ….. (1)

    We know that

    a, c, a2 + b2 and b2 + c2 are in proportion

    It can be written as

    a/c = (a2 + b2)/ (b2 + c2)

    By cross multiplication

    a (b2 + c2) = c (a2 + b2)

    Using equation (1)

    a (ac + c2) = c (a2 + ac)

    So we get

    ac (a + c) = a2c + ac2

    Here ac (a + c) = ac (a + c) which is true.

    Therefore, it is proved.

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