[list=1][*]To find the lengths of the sides of the squares, take the square root of the area, [math]l=\sqrt{A}[/math]. [b]Square 1 -[/b] [math]l=\sqrt{4}=\sqrt{2^2}=2[/math]. [b]Square 2 -[/b] [math]l=\sqrt{49}=\sqrt{7^2}=7[/math].[/*][*]To find the lengths of the [b][color=#6aa84f]legs[/color][/b] of the right triangles, take the square root of the area of the square, [math]l=\sqrt{A}[/math]. To estimate the length of the [color=#6aa84f][b]hypotenuse[/b][/color], know that it is greater than either of the [b][color=#6aa84f]leg[/color][/b] lengths, but less than both [color=#6aa84f][b]leg[/b][/color] lengths added together, [i]l[sub]1[/sub] or l[sub]2[/sub] < h <[/i][i] l[sub]1[/sub]+l[sub]2[/sub] [/i]. [b]Right Triangle 3 -[/b] [math]l=\sqrt{100}=\sqrt{10^2}=10[/math] & [i]h btwn 10 to 20. [/i][b]Right Triangle 4 -[/b][i] [/i][i][/i][math]l=\sqrt{25}=\sqrt{5^2}=5[/math][i] & h btwn 5 to 10.[/i][/*][/list]

[b]Version 1:[/b] Compose the fixed square using 4 copies of the same right triangle.[br][list=1][*][math]A=4\left(\frac{1}{2}\cdot36\right)=4\left(18\right)=72[/math] units[sup]2[/sup] [/*][*][math]l=\sqrt{72}\cong8.49[/math] units[sup]2[/sup][/*][/list][br][b]Version 2:[/b] Compose the fixed square using 1 large square and subtracting 4 copies of the same right triangle.[br][list=1][*][math]A=144-4\left(\frac{1}{2}\cdot36\right)=144-4\left(18\right)=144-72=72[/math] units[sup]2[/sup][/*][*][math]l=\sqrt{72}\cong8.49[/math] units[sup]2[/sup][/*][/list]

Compose the inner square using the outer square and subtracting 4 copies of the same right triangle.[br][br]Find the area of the inner square.[br][list][*][math]A=12^2-4\left[\frac{1}{2}\left(4\cdot8\right)\right]=144-4\left[\frac{1}{2}\left(32\right)\right]=144-4\left[16\right]=144-64=80[/math] units[sup]2[/sup][/*][/list][br]Find the side length of the inner square (this will also be the [color=#6aa84f][b]hypotenuse[/b][/color] length).[br][list][*][math]l=\sqrt{80}\cong8.94[/math] units[/*][/list]

[list=1][*]Compose the square by taking a larger square and subtracting 4 same right triangles. [math]A=4^2-4\left[\frac{1}{2}\left(2^2\right)\right]=16-4\left[\frac{1}{2}\left(4\right)\right]=16-4\left[2\right]=16-8=8[/math], therefore [math]l=\sqrt{8}\cong2.83[/math]. The distance between (2, 2) and (4, 4) is about 2.83 units.[/*][*]Compose the square by taking 4 same right triangles and adding a smaller square. [math]A=4\left[\frac{1}{2}\left(10\cdot5\right)\right]+5^2=4\left[\frac{1}{2}\left(50\right)\right]+25=4\left[25\right]+25=100+25=125[/math], therefore [math]l=\sqrt{125}\cong11.18[/math]. The distance between (-4, 6) and (1, -4) is about 11.18 units.[/*][/list]