Distance Across I Don't Know Where to Begin

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Discussion Overview

The discussion revolves around a mathematical problem involving the dimensions of segments represented by tick marks and the calculation of the diagonal of a rectangle formed by these segments. Participants explore the relationships between the lengths of the segments and their implications for the area and diagonal length of the rectangle.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant defines the lengths of segments with tick marks as $a$ and $b$, leading to the equation $a^2+b^2=50$.
  • Another participant suggests the dimensions of the rectangle are $\sqrt{2}a$ and $\sqrt{2}b$.
  • A correction is made regarding the dimensions, stating they should be $\sqrt{2}a$ and $\sqrt{2}b$, prompting a question about the diagonal of the rectangle.
  • Participants apply the Pythagorean theorem to express the diagonal length as $\overline{MK}=\sqrt{2(a^2+b^2)}$.
  • There is a calculation presented that leads to the conclusion that the length of $\overline{MK}$ is 10, based on the earlier established equation.

Areas of Agreement / Disagreement

The discussion shows some agreement on the application of the Pythagorean theorem and the resulting calculations, but there are also corrections and clarifications regarding the dimensions of the rectangle and the diagonal that indicate some uncertainty in the initial statements.

Contextual Notes

Participants rely on the assumption that $a^2+b^2=50$ without further clarification on the context of this equation. The discussion does not resolve potential ambiguities in the definitions of $a$ and $b$ or the implications of the calculations presented.

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View attachment 6484 I don't know where to begin
 

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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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