1.
[tex] {y}^{2} = {40}^{2} + {20}^{2} \\ {y}^{2} = 1600 + 400 \\ {y}^{2} = 2000 \\ y = \sqrt{2000} \\ y = \sqrt{400 \times 5} \\ y = 20 \sqrt{5} [/tex]
2.
[tex] \frac{1}{2} a \sqrt{2} = \frac{1}{2} \times 8 \sqrt{2} = 4 \sqrt{2} [/tex]
[tex] {x}^{2} + (4 \sqrt{2} {)}^{2} = {10}^{2} \\ {x}^{2} + 16 \times 2 = 100 \\ {x}^{2} + 32 = 100 \\ x = \sqrt{68} \\ x = \sqrt{4 \times 17} \\ x = 2 \sqrt{17} [/tex]
3.
[tex] \frac{2}{3} h = \frac{2}{3} \times \frac{a \sqrt{3} }{2} = \frac{2}{3} \times \frac{4 \sqrt{3} \times \sqrt{3} }{2} = \frac{2}{3} \times \frac{4 \sqrt{9} }{2} = \frac{2}{3} \times \frac{4 \times 3}{2} = \frac{2}{3} \times \frac{12}{2} = \frac{2}{3} \times 6 = \frac{12}{3} = 4[/tex]
[tex] {x}^{2} = {10}^{2} + {4}^{2} \\ {x}^{2} = 100 + 16 \\ x = \sqrt{116} \\ x = \sqrt{4 \times 29} \\ x = 2 \sqrt{29} [/tex]
4.
[tex] \frac{2}{3} h = \frac{2}{3} \times \frac{a \sqrt{3} }{2} = \frac{2}{3} \times \frac{6 \sqrt{3} \times \sqrt{3} }{2} = \frac{2}{3} \times \frac{6 \sqrt{9} }{2} = \frac{2}{3} \times \frac{6 \times 3}{2} = \frac{2}{3} \times \frac{18}{2} = \frac{2}{3} \times 9 = \frac{18}{3} = 6[/tex]
[tex] {y}^{2} + {6}^{2} = {8}^{2} \\ {y}^{2} + 36 = 64 \\ {y}^{2} = 28 \\ y = \sqrt{28} \\ y = \sqrt{4 \times 7} \\ y = 2 \sqrt{7} [/tex]
