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In the above figure, seg PA, seg QB and ...


In the above figure, seg PA, seg QB and seg RC are perpendicular to seg AC. From the information given in the figure, prove that:
`(1)/(x) + (1)/(z) = (1)/(y)`

Text Solution

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In `Delta PAC` and `Delta QBC`
`/_PAC ~= /_ QBC`….(given)
`/_ PCA ~= /_QCB`….(Common angle)
`:. Delta PAC ~ Delta QBC` …..(By AA test)
`:. (PA)/(QB) = (AC)/(BC)`….(Corresponding sides of similar tirangles are in proportion)
`(x)/(y) = (AC)/(BC)`
`(y)/(x) = (BC)/(AC)`...(I) ...(by invertendo)
Similarly, `Delta RCA ~ Delta QBA`..... (By AA test)
`(RC)/(QB) = (AC)/(AB)`...(Corresponding sides of similar triangles are in proportion
`:. (z)/(y) = (AC)/(AB)`
`:. (y)/(z) = (AB)/(AC)`.....(II)....(by invertendo)
`:. (y)/(x) + (y)/(z) = (BC)/(AC) + (AB)/(AC)` [Adding (I) and (II)]
`:. y ((1)/(x) + (1)/(Z)) = (BC + AB)/(AC)`
`:. y ((1)/(x) + (1)/(z)) = (AC)/(AC) (A-B-C)`
`:. ((1)/(x) + (1)/(z)) = 1`
`:. (1)/(x) + (1)/(z) = (1)/(y)`
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Knowledge Check

  • In the given figure, if PA = x, RC = y and QB = z, then which one of the following is correct?

    A
    `2y = x+z `
    B
    `4y = x + z `
    C
    `xy + yz= xz`
    D
    `xy+xz=yz`