Distance Across I Don't Know Where to Begin

[Ilikebugs]

View attachment 6484 I don't know where to begin

 

[MarkFL]

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:

\(\displaystyle a^2+b^2=50\)

In terms of $a$ and $b$, what are the dimensions of the white rectangle within the tile?

 

[Ilikebugs]

√2a^2 and √2b^2

 

[MarkFL]

√2a^2 and √2b^2

Not quite...it would be \(\displaystyle \sqrt{2}a\) and \(\displaystyle \sqrt{2}b\)...so what would the diagonal of the rectangle be?

 

[Ilikebugs]

sqr(2a+2b) ?

 

[MarkFL]

sqr(2a+2b) ?

Using the Pythagorean theorem, we find:

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

Now, we know that $a^2+b^2=50$, so what is the length of $\overline{MK}$?

 

[Ilikebugs]

|MK|=√(a√2)^2+(b√2)^2?

 

[Ilikebugs]

|MK|=√(a√2)^2+(b√2)^2?

MK equals 100?

 

[MarkFL]

MK equals 100?

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