Zadanie 1
1)
a=3+2\sqrt2
c=6+\sqrt2
{x=\sqrt{c^2-a^2}=\sqrt{(6+\sqrt2)^2-(3+2\sqrt2)^2}=\sqrt{36+12\sqrt2+2-(9+12\sqrt2+4*2)}=}
=\sqrt{38+12\sqrt2-17-12\sqrt2}=\sqrt{21}
2)
a=2x
b=\frac{4}{3}\sqrt2
c=\sqrt6
(\sqrt6x)^2=(2x)^2+(\frac{4}{3}\sqrt2)^2 , x>0
6x^2=4x^2+\frac{16*2}{9}
2x^2=\frac{32}{9} \ |:2
x^2=\frac{16}{9}
x=\frac{4}{3}
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a=2x=2*\frac{4}{3}=\frac{8}{3}=2\frac{1}{3}
c=\sqrt6x=\frac{4}{3}*\sqrt6=\frac{4\sqrt3}{6}
3)
h=\sqrt{7^2-(\sqrt{37})^2}=\sqrt{49-37}=\sqrt{12}=\sqrt{4*3}=2\sqrt{3}
b=\sqrt{2^2+(2\sqrt3)^2}=\sqrt{4+12}=\sqrt{16}=4
4)
a=6, b=2
c=\sqrt{6^2+2^2}=\sqrt{36+4}=\sqrt{40}=\sqrt{4*10}=2\sqrt{10}
5)
a=3, b=5
c=\sqrt{3^2+5^2}=\sqrt{9+25}=\sqrt{34}
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a=4, c=3\sqrt2
b=\sqrt{(3\sqrt2)^2-4^2}=\sqrt{9*2-16}=\sqrt2
6)
a=2, b=2\sqrt3
c=\sqrt{2^2+(2\sqrt3)^2}=\sqrt{4+4*3}=\sqrt{16}=4
7)
a=3, b=3\sqrt2
c=\sqrt{3^2+(3\sqrt2)^2}=\sqrt{9+9*2}=\sqrt{27}=\sqrt{9*3}=3\sqrt3
8)
a=5, b=6
c=\sqrt{5^2+6^2}=\sqrt{25+36}=\sqrt{61}
9)
a=8, c=12
b=\sqrt{12^2-8^2}=\sqrt{144-64}=\sqrt{80}=\sqrt{16*5}=4\sqrt5
10)
a=\sqrt{11} c=5
b=\sqrt{5^2-(\sqrt{11})^2}=\sqrt{25-11}=\sqrt{14}