trojkaty prostokatne

[dusiek91]

oblicz pole rownoramiennego trojkata prostokatnego , gdy;
a) jego obwob jest rowny \(\displaystyle{ 6(1+ \sqrt{2} )}\)
b) jego przeciwprostokatna jest o \(\displaystyle{ 1+ \sqrt{2}}\) dluzsza od przyprostokatnej

Pamiętaj o klamrach \(\displaystyle{ !
luka52}\)

[raphel]

a)
\(\displaystyle{ \begin{cases} 2a+c = 6+6 \sqrt{2} \\ 2a ^{2} = c ^{2} \end{cases}}\)
\(\displaystyle{ \begin{cases} 2a+c = 6+6 \sqrt{2} \\ c= \sqrt{2}a \end{cases}}\)
\(\displaystyle{ 2a + \sqrt{2} a = 6 + 6 \sqrt{2}}\)
\(\displaystyle{ (2+ \sqrt{2} )a = 6 + 6 \sqrt{2}}\)
\(\displaystyle{ a = \frac{6 + 6 \sqrt{2}}{2+ \sqrt{2}} \frac{2- \sqrt{2}}{2- \sqrt{2}}}\)
\(\displaystyle{ a = 3 \sqrt{2}}\)
\(\displaystyle{ c=6}\)
b)
\(\displaystyle{ \begin{cases} c=1+ \sqrt{2} + a \\ 2a ^{2} = c ^{2} \end{cases}}\)
\(\displaystyle{ c= \sqrt{2}a}\)
\(\displaystyle{ \sqrt{2}a - a = 1+ \sqrt{2}}\)
\(\displaystyle{ a = \frac{1+ \sqrt{2} }{ \sqrt{2} -1 }}\)
\(\displaystyle{ a = 3+2 \sqrt{2}}\)
\(\displaystyle{ c = 4+3 \sqrt{2}}\)

a pola to już łatwo obliczyć;)