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May 30th, 2016, 02:35 PM   #1
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how to find length MC in a rectangle

length MC.pdf

Please click the link above to see a diagram.

The diagram shows a rectangle with vertices ABCD. M is a point inside the rectangle. AM is 3cm, BM is 12cm and DM is 11cm. Find the length of MC using coordinate geometry.

Can someone please explain how to do it? Thank you very much.
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May 30th, 2016, 08:16 PM   #2
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Express the squares of the distances in terms of the coordinates by use of Pythagoras.

MC = 16cm (because 11² + 12² = 3² + 16²)
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May 30th, 2016, 08:20 PM   #3
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Let $\displaystyle D = (0,0)$, $\displaystyle M = (h, k)$, $\displaystyle B = (p, q)$. Then

$\displaystyle
\begin{cases} h^2 + k^2 = 11^2 = 121 & ...(1) \\ h^2 + (q-k)^2 = 3^2 = 9& ...(2)\\ (p-h)^2 + (q-k)^2 = 12^2 = 144 & ...(3)\end{cases}$

And we need to find $\displaystyle \sqrt{(p-h)^2 + k^2}$.

$\displaystyle (1) - (2):\\
k^2 - (q-k)^2 = 112 ...(4) $

$\displaystyle (4) + (3):\\
(p-h)^2 + k^2 = 256 = 16^2$

Thus $\displaystyle MC = 16$.
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