## Task 3 - 030913

a) Construct a triangle from $a=5.6 \mathrm{~cm}, r=3.5 \mathrm{~cm}$ (radius of the circumcircle) and $\gamma=60^{\circ}$!

b) Describe the construction!

c) Calculate the distance of the circumcenter from the side $a$!

d) Investigate for which measures of the circumcircle radius the construction of a triangle with $a=$ $5.6 \mathrm{~cm}$ and $\gamma=60^{\circ}$ is not possible!
a) See figure.

b) Start with side $a$ and label its endpoints with $B$ and $C$. Then draw circular arcs with radius $r$ around the two endpoints of $a$, so that an intersection point is created. This point is then the circumcenter of the desired triangle; let it be called $M$.

Then construct the circle around $M$ with radius $r$, on which point $A$ must lie. The latter is created when the angle $\gamma$ at $a$ in $C$ is set such that $\gamma$ is an interior angle of the resulting triangle $A B C$.

c) The triangle $B C M$ is isosceles. From the Pythagorean theorem, the height on $a$, which is also the desired distance $d$, follows: $d=\sqrt{r^{2}-(a / 2)^{2}}=2.1 \text{ cm}$.

d) The triangle $B C M$ must exist, i.e., its triangle inequality must be satisfied: $2 r \geq a$ or $r \geq 2.8 \text{ cm}$.