B3. Given the points $A(5,5), B(-2,4)$, and $C(-1,-3)$. Calculate the area of triangle $ABC$ and the length of its shortest altitude.
B3. The area of a triangle is equal to half the absolute value of the determinant. We insert the data into the formula $S=\frac{|D|}{2}$. The determinant is equal to 50, so the area is 25. We calculate the lengths of the sides of the triangle, which measure $|A C|=10$, $|B C|=5 \sqrt{2}$, and $|A B|=5 \sqrt{2}$ units. The shortest height is the height to the longest side. We calculate it from the area $v_{A C}=\frac{2 S}{|A C|}$. The height measures 5 units.

We can solve the problem differently. We calculate the area of the trapezoid $F A B E$, which has bases 8 and 7 units long and a height of 7 units (see the figure). This is equal to $\frac{8+7}{2} \cdot 7=\frac{15 \cdot 7}{2}$. The area of triangle $A B C$ is obtained by subtracting the areas of the right triangles with legs $|B E|=7$ and $|E C|=1$ as well as $|A F|=8$ and $|C F|=6$. The area of the triangle is therefore $\frac{15 \cdot 7}{2}-\frac{1 \cdot 7}{2}-\frac{8 \cdot 6}{2}=\frac{14 \cdot 7}{2}-24=25$. The longest side of triangle $A B C$ is $A C$, which is $\sqrt{6^{2}+8^{2}}=\sqrt{100}=10$ units long. The shortest height is therefore $v_{A C}=\frac{2 \cdot 25}{10}=5$.

Knowledge of the formula for the determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-11.jpg?height=69&width=1641&top_left_y=1693&top_left_x=276)

![](https://cdn.mathpix.com/cropped/2024_06_07_8d4f73ada7862f989e67g-11.jpg?height=52&width=1639&top_left_y=1756&top_left_x=274)

Use of the formula for the length of a segment (distance between points) ............................................................

Calculation of the lengths of the sides of the triangle........................................................................................................

Calculated length of the height to side AC....................................................................