Source Investigations

Hypotenuses: 3$\sqrt{2}$, 4$\sqrt{2}$, 6$\sqrt{2}$, 8$\sqrt{2}$, 9$\sqrt{2}$. One-decimal bounds: $\left(4.2,4.3\right)$, $\left(5.6,5.7\right)$, $\left(8.4,8.5\right)$, $\left(11.3,11.4\right)$, $\left(12.7,12.8\right)$. Square the endpoints to verify each strict bound.

Isosceles right hypotenuse 10 gives each leg 5$\sqrt{2}$. Page 47 Q4 answers: $\sqrt{74}$, 4$\sqrt{13}$, 12, $\sqrt{193}$, $\sqrt{58}$/2. For the final case, 1.$5^2$+3.$5^2$=14.5=$\frac{58}{4}$.

Combine a square of area A with its doubled square of area 2A to obtain 3A. Double the doubled square to obtain 4A, then combine with A to obtain 5A. The required sides are $\sqrt{3A}$ and $\sqrt{5A}$.

Positive legs give $c^2$=$a^2$+$b^2$>$a^2$ and $c^2$>$b^2$. Taking positive roots gives c>a and c>b.

Rectangles 3×4, 5×12, 8×15, 7×24 and 12×35 have diagonals 5,13,17,25,37 respectively. Divide every entry of any non-primitive triple by its gcd: dividing the squared equality by that gcd squared yields a primitive triple.