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Question Number 170202 by otchereabdullai@gmail.com last updated on 18/May/22

A point is 3 cm, 4 cm and 5 cm away

from three vertices of a rectangle .

How far can it be from the 4 th vertex .

Find all solutions

Commented by otchereabdullai@gmail.com last updated on 18/May/22

much much grateful thanks

Commented by mr W last updated on 18/May/22

t h e r e a r e i n f i n i t e l y m a n y s o l u t i o n s .

Commented by MJS_new last updated on 18/May/22

{it}^{'} s either 0 \lor 3 \sqrt{2} \lor 4 \sqrt{2}

Commented by otchereabdullai@gmail.com last updated on 18/May/22

please help me with some of the

solutions prof W

Commented by otchereabdullai@gmail.com last updated on 18/May/22

prof mjs how did you get the

0 \lor 3 \sqrt{2} \lor 4 \sqrt{2}

Commented by MJS_new last updated on 18/May/22

“ {fix}^{″} the vertices

A = (\begin{matrix} 0 \\ 0 \end{matrix}) B = (\begin{matrix} a \\ 0 \end{matrix}) C = (\begin{matrix} a \\ b \end{matrix}) D = (\begin{matrix} 0 \\ b \end{matrix})

let P = (\begin{matrix} p \\ q \end{matrix})

equations with α, β, γ unknown at first

(1) ∣ A P ∣^{2} = α^{2}

(2) ∣ B P ∣^{2} = β^{2}

(3) ∣ C P ∣^{2} = γ^{2}

(4) ∣ D P ∣^{2} = x^{2}

you can easily solve the system for p, q and

get a formula for x^{2} . now insert all possible

permutations of 3, 4, 5 for α, β, γ

Commented by mr W last updated on 18/May/22

y o u a r e r i g h t s i r!

t h o u g h t h e r e a r e i n f i n i t e l y m a n y s u c h

r e c t a n g l e s, b u t t h e d i s t a n c e t o t h e

f o u r t h v e r t e x i s c o n s t a n t .

x = \sqrt{α^{2} + β^{2} - γ^{2}}

x = \sqrt{3^{2} + 4^{2} - 5^{2}} = 0

x = \sqrt{3^{2} + 5^{2} - 4^{2}} = 3 \sqrt{2}

x = \sqrt{4^{2} + 5^{2} - 3^{2}} = 4 \sqrt{2}

Commented by mr W last updated on 18/May/22

Commented by mr W last updated on 18/May/22

α^{2} = a_{1}^{2} + b_{2}^{2}

β^{2} = a_{2}^{2} + b_{1}^{2}

\Rightarrow α^{2} + β^{2} = a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}

x^{2} = a_{2}^{2} + b_{2}^{2}

γ^{2} = a_{1}^{2} + b_{1}^{2}

\Rightarrow x^{2} + γ^{2} = a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}

\Rightarrow x^{2} + γ^{2} = α^{2} + β^{2}

Commented by Tawa11 last updated on 08/Oct/22

Great sirs

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