B2 Points $A(-4,-2), B(2,-2)$ and $C(2,6)$ are the vertices of triangle $A B C$.

A Draw triangle $A B C$ in the coordinate system.

B Calculate the perimeter of triangle $A B C$.

C Calculate the area of triangle $A B C$.

D Circumscribe a circle around triangle $A B C$.
B2 Points $A(-4,-2), B(2,-2)$ and $C(2,6)$ are plotted in a Cartesian coordinate system and connected to form a right triangle $\triangle A B C$ with a right angle at vertex $C$.

![](https://cdn.mathpix.com/cropped/2024_06_07_8c4ca9fb11cdb9201aa3g-6.jpg?height=994&width=860&top_left_y=905&top_left_x=198)

The lengths of the legs are $|A B|=6 e$ and $|B C|=8 e$, and the length of the hypotenuse $|A C|$ can be found using the Pythagorean theorem: $|A C|^{2}=|A B|^{2}+|B C|^{2}$. Thus, $|A C|^{2}=(6 e)^{2}+(8 e)^{2}$, from which we get $|A C|=10 e$. The perimeter of $\triangle A B C$ is $|A B|+|B C|+|A C|=6 e+8 e+10 e=24 e$. The area of the triangle $\triangle A B C$ is $p_{\triangle A B C}=\frac{|A B| \cdot|B C|}{2}=\frac{6 \cdot 8}{2}=24 e^{2}$. The circumcircle of the right triangle has its center at the midpoint of the hypotenuse, and the radius is half the length of the hypotenuse.

A The plotted and connected points. ....................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_8c4ca9fb11cdb9201aa3g-6.jpg?height=49&width=1650&top_left_y=2397&top_left_x=266)

Calculated perimeter of the triangle 24 e. .......................................................... 1 t

C Calculated area of the triangle $24 e^{2}$. ..............................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_8c4ca9fb11cdb9201aa3g-6.jpg?height=52&width=1662&top_left_y=2624&top_left_x=254)

Note: If the contestant does not include units in the results, we deduct 1 point from the total score.