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प्रश्न
Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.
उत्तर
Let ABCD be a quadrilateral in which
`|bar"AB"| = |bar"BC"| = |bar"CD"| = |bar"DA"|` ....(1)
and AB || DC and AD || BC
∴ `bar"AB" = bar"DC"` and `bar"AD" = bar"BC"` ...(2)
Now, `bar"AC" = bar"AB" + bar"BC"`
and `bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"BC"` ...[By(2)]
`= bar"BC" - bar"AB"`
∴ `bar"AC".bar"BD" = (bar"AB" + bar"BC").(bar"BC" - bar"AB")`
`= bar"AB".(bar"BC" - bar"AB") + bar"BC" . (bar"BC" - bar"AB")`
`= bar"AB".bar"BC" - bar"AB".bar"AB" + bar"BC".bar"BC" - bar"BC".bar"AB"`
`= |bar"BC"|^2 - |bar"AB"|^2` ....`[∵ bar"AB".bar"BC" = bar"BC".bar"AB"]`
= 0 ...[By(1)]
∵ `bar"AC", bar"BD"` are non-zero vectors
∴ `bar"AC"` is perpendicular to `bar"BD"`
Hence, the diagonals are perpendicular.
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