Question

Angle b=angle f = angle d=90 ab=x, cd=z, ef=y then prove tht 1/x+1/y=1/z?

Answer 1

The given result is incorrect and the correct relationship is:

`1/y = 1/z + 1/x`

Explanation:

`triangle ABC` and `triangle DEF` are similar, ie `AB:EF = BD:DF`

`:. x/y = (BD)/(DF)`
`:. DF = (BD) y/x`

`triangle BCD` and `triangle BEF` are similar, ie `CD:EF = BD:BF`

`:. z/y = (BD)/(BF)`
`:. BF = (BD) y/z`

Note that `BD=BF+DF`, and so:

`BD = (BD) y/z + (BD) y/x`

If we cancel the dimension `(BD)` we get

`1 = y/z + y/x`

And dividing by `y` we get:

`1/y = 1/z + 1/x`

And the given result is incorrect:

VERIFICATION EXAMPLE

We can see this with a simple example:

In this example, we have:

`x=6, y=2, z=3`

The result quote in the question would have:

`1/z=1/x+1/y`
`1/z=1/3`, and `1/x+1/y =1/6+1/2=2/3`

Showing the quoted result is invalid.

However if we use the derived result:

`1/y = 1/z + 1/x`
`1/y=1/2`, and `1/z+1/x=1/3+1/6=1/2`