1997 Paper 1 Q5

Year: 1997
Paper: 1
Question Number: 5

Course: LFM Pure and Mechanics
Section: Vectors

Difficulty: 1500.0 Banger: 1484.0

vectors geometric proof quadrilateral midpoints distance formula scalar product invariant

Problem

Four rigid rods \(AB\), \(BC\), \(CD\) and \(DA\) are freely jointed together to form a quadrilateral in the plane. Show that if \(P\), \(Q\), \(R\), \(S\) are the mid-points of the sides \(AB\), \(BC\), \(CD\), \(DA\), respectively, then \[|AB|^{2}+|CD|^{2}+2|PR|^{2}=|AD|^{2}+|BC|^{2}+2|QS|^{2}.\] Deduce that \(|PR|^{2}-|QS|^{2}\) remains constant however the vertices move. (Here \(|PR|\) denotes the length of \(PR\).)

No solution available for this problem.

Rating Information

Difficulty Rating: 1500.0

Difficulty Comparisons: 0

Banger Rating: 1484.0

Banger Comparisons: 1

Show LaTeX source

Problem source

Four rigid rods $AB$, $BC$, $CD$ and $DA$
are freely jointed together to form a quadrilateral
in the plane.
Show
that if $P$, $Q$, $R$, $S$ are the mid-points of the sides $AB$,
$BC$, $CD$, $DA$, respectively, then
\[|AB|^{2}+|CD|^{2}+2|PR|^{2}=|AD|^{2}+|BC|^{2}+2|QS|^{2}.\]
Deduce that $|PR|^{2}-|QS|^{2}$
remains constant however the vertices move. (Here $|PR|$
denotes the length of $PR$.)

Back to List