Question

logical reasoning the quilt design in the photo is based on the pattern in the diagram below. use the diagram in exercises 31–34. wheel of theodorus 31. find the values of r, s, t, u, v, and w. explain the procedure you used to find the values.

Explanation:

Step1: Calculate r via Pythagoras

Each triangle is right-angled, with legs 1 and the prior hypotenuse. For the first triangle:
$r = \sqrt{1^2 + 1^2} = \sqrt{2}$

Step2: Calculate s via Pythagoras

Use leg 1 and hypotenuse $r$:
$s = \sqrt{1^2 + (\sqrt{2})^2} = \sqrt{1+2} = \sqrt{3}$

Step3: Calculate t via Pythagoras

Use leg 1 and hypotenuse $s$:
$t = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1+3} = \sqrt{4} = 2$

Step4: Calculate u via Pythagoras

Use leg 1 and hypotenuse $t$:
$u = \sqrt{1^2 + 2^2} = \sqrt{1+4} = \sqrt{5}$

Step5: Calculate v via Pythagoras

Use leg 1 and hypotenuse $u$:
$v = \sqrt{1^2 + (\sqrt{5})^2} = \sqrt{1+5} = \sqrt{6}$

Step6: Calculate w via Pythagoras

Use leg 1 and hypotenuse $v$:
$w = \sqrt{1^2 + (\sqrt{6})^2} = \sqrt{1+6} = \sqrt{7}$

Answer:

$r=\sqrt{2}$, $s=\sqrt{3}$, $t=2$, $u=\sqrt{5}$, $v=\sqrt{6}$, $w=\sqrt{7}$