Distance Across I Don't Know Where to Begin

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SUMMARY

The discussion focuses on calculating the length of the diagonal of a rectangle formed by segments with tick marks, defined by lengths \(a\) and \(b\). The relationship \(a^2 + b^2 = 50\) is established, leading to the conclusion that the diagonal length \(\overline{MK}\) is derived using the Pythagorean theorem. The final calculation shows that \(\overline{MK} = 10\) cm, confirming the dimensions of the rectangle based on the given parameters.

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I would let $a$ be the length (in cm) of the segments with 1 tick mark, and $b$ be the length (in cm) of the segments with 2 tick marks. And so, given the statement regarding the area of the colored sections, we may write:

$$a^2+b^2=50$$

In terms of $a$ and $b$, what are the dimensions of the white rectangle within the tile?
 
√2a^2 and √2b^2
 
Ilikebugs said:
√2a^2 and √2b^2

Not quite...it would be $$\sqrt{2}a$$ and $$\sqrt{2}b$$...so what would the diagonal of the rectangle be?
 
sqr(2a+2b) ?
 
Ilikebugs said:
sqr(2a+2b) ?

Using the Pythagorean theorem, we find:

$$\overline{MK}=\sqrt{(\sqrt{2}a)^2+(\sqrt{2}b)^2}=\sqrt{2\left(a^2+b^2\right)}$$

Now, we know that $a^2+b^2=50$, so what is the length of $\overline{MK}$?
 
|MK|=√(a√2)^2+(b√2)^2?
 
Ilikebugs said:
|MK|=√(a√2)^2+(b√2)^2?
MK equals 100?
 
Ilikebugs said:
MK equals 100?

$$\overline{MK}=\sqrt{(\sqrt{2}a)^2+(\sqrt{2}b)^2}=\sqrt{2\left(a^2+b^2\right)}=\sqrt{2(50)}=\sqrt{100}=10$$ :D
 

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