1.) Lösung folgt noch...
2.) Für die Lösung muss man sich ein gleichschenkeliges Dreieck vorstellen.
a
=
b
,
α
=
β
{\displaystyle a=b,\quad \alpha =\beta }
tan
α
=
G
e
g
e
n
k
a
t
h
e
t
e
A
n
k
a
t
h
e
t
e
=
30
10
=
3
⇒
α
=
β
≈
71
,
565
∘
{\displaystyle \tan \ \alpha \ =\ {\frac {Gegenkathete}{Ankathete}}\ =\ {\frac {30}{10}}\ =\ 3\quad \Rightarrow \quad \alpha =\beta \approx 71,565^{\circ }}
γ
=
180
−
(
α
+
β
)
≈
36
,
87
∘
{\displaystyle \gamma =180-(\alpha +\beta )\approx 36,87^{\circ }}
c
=
10
2
+
30
2
≈
31
,
62
c
m
{\displaystyle c={\sqrt {10^{2}+30^{2}}}\approx 31,62cm}
a
sin
α
=
c
sin
γ
⇔
a
=
c
sin
γ
⋅
sin
α
⇒
a
=
50
c
m
{\displaystyle {\frac {a}{\sin \alpha }}={\frac {c}{\sin \gamma }}\quad \Leftrightarrow \quad a={\frac {c}{\sin \gamma }}\cdot \sin \alpha \quad \Rightarrow \quad a=50cm}
Der See ist 40 cm tief .
3.) a)
c
2
=
a
2
+
b
2
{\displaystyle c^{2}=a^{2}+b^{2}\,}
(
p
+
q
)
2
=
h
2
+
p
2
+
h
2
+
q
2
{\displaystyle (p+q)^{2}=h^{2}+p^{2}+h^{2}+q^{2}\,}
p
2
+
2
p
q
+
q
2
=
2
h
2
+
p
2
+
q
2
{\displaystyle p^{2}+2pq+q^{2}=2h^{2}+p^{2}+q^{2}\,}
2
h
2
=
2
p
q
{\displaystyle 2h^{2}=2pq\,}
h
2
=
p
⋅
q
{\displaystyle h^{2}=p\cdot q}
b)
c
2
=
a
2
+
b
2
{\displaystyle c^{2}=a^{2}+b^{2}\,}
c
(
p
+
q
)
=
a
2
+
b
2
{\displaystyle c(p+q)=a^{2}+b^{2}\,}
c
⋅
p
+
c
⋅
q
=
a
2
+
b
2
|
−
b
2
{\displaystyle c\cdot p+c\cdot q=a^{2}+b^{2}\quad \vert -b^{2}}
a
2
=
c
p
+
c
q
−
(
h
2
+
q
2
)
{\displaystyle a^{2}=cp+cq-(h^{2}+q^{2})\,}
a
2
=
c
p
+
c
q
−
p
q
−
q
2
{\displaystyle a^{2}=cp+cq-pq-q^{2}\,}
a
2
=
(
c
−
q
)
⋅
(
p
+
q
)
{\displaystyle a^{2}=(c-q)\cdot (p+q)\,}
a
2
=
p
⋅
c
_
{\displaystyle {\underline {a^{2}=p\cdot c}}}
c
⋅
p
+
c
⋅
q
=
a
2
+
b
2
|
−
a
2
{\displaystyle c\cdot p+c\cdot q=a^{2}+b^{2}\quad \vert -a^{2}}
b
2
=
q
⋅
c
_
{\displaystyle {\underline {b^{2}=q\cdot c}}}
h
2
=
p
q
{\displaystyle h^{2}=pq\,}
h
2
=
(
c
−
q
)
(
c
−
p
)
{\displaystyle h^{2}=(c-q)(c-p)\,}
h
2
=
c
2
−
p
c
−
q
c
+
p
q
|
−
h
2
,
+
p
c
,
+
q
c
{\displaystyle h^{2}=c^{2}-pc-qc+pq\quad \vert -h^{2},+pc,+qc}
c
2
=
p
c
+
q
c
=
a
2
+
b
2
{\displaystyle c^{2}=pc+qc=a^{2}+b^{2}\,}
