24. Calculate the area of a pentagon whose vertices have the coordinates: $(-1 ; 2),(1 ; 1)$, $(2 ; 3),(3 ;-5)$ and $(-2 ;-4)$.
Solution. Let's make a "rough" drawing (Fig. 10). We have:

![](https://cdn.mathpix.com/cropped/2024_05_21_b1a79769c6c63d19f37fg-18.jpg?height=302&width=231&top_left_y=523&top_left_x=570)

![](https://cdn.mathpix.com/cropped/2024_05_21_b1a79769c6c63d19f37fg-18.jpg?height=545&width=645&top_left_y=230&top_left_x=1071)

Fig. 10

From this, \( F_{3}=((2+4+10+9+2)-(-1-4-12-10+3))=51 \). Thus, \( S_{5}=\frac{1}{2} \cdot|51|=25.5 \).

## DETERMINING THE CONVEXITY OF FIGURES ON A COORDINATE PLANE USING THE COORDINATE METHOD

A set of points is called convex if, together with any two of its points \( M_{1} \) and \( M_{2} \), it also contains all points of the segment \( M_{1} M_{2} \). Examples of convex sets include a half-plane, a circle, a triangle, and others. Polygons can be either convex or non-convex. Whether a figure is convex can be determined using geometric and analytical (coordinate) methods. Let's focus on the coordinate method.

The equation of a line passing through two given points \( M_{1}\left(x_{i} ; y_{1}\right) \) and \( M_{2}\left(x_{2} ; y_{2}\right) \) is given by (see problem 23):

\[
\left(x-x_{1}\right)\left(y_{2}-y_{1}\right)=\left(y-y_{1}\right)\left(x_{2}-x_{1}\right)
\]

or, when \( x_{1} \neq x_{2} \) and \( y_{1} \neq y_{2} \),

\[
\frac{x-x_{i}}{x_{2}-x_{i}}=\frac{y-y_{i}}{y_{2}-y_{i}}
\]

Let \(\frac{x-x_{i}}{x_{2}-x_{1}}=\frac{y-y_{i}}{y_{2}-y_{i}}=k\), then

\[
\left\{\begin{array}{l}
x=(1-k) x_{1}+k x_{2} \\
y=(1-k) y_{i}+k y_{2}
\end{array}\right.
\]

Equations ( \(1^{\prime}\) ) and (1) represent the parametric equations of the line \( M_{1} M_{2} \) with the current point \( M(x ; y) \).

Remark. Equation (1) is the equation of the line \( M_{1} M_{2} \), which is not parallel to the coordinate axes. If this line is parallel to the coordinate axes, its equation will be:

\[
x=x_{i} \quad \text { or } \quad y=y_{i} .
\]

Task 1. By making the appropriate drawings, determine the following for the parameter \( k \):

a) If the point \( M \) lies on the line \( M_{1} M_{2} \) and the vectors \( \overline{M_{1} M_{2}} \) and \( \overline{M_{1} M} \) are directed in the same way, then \( k>0 \); if these vectors are directed oppositely, then \( k<0 \);

b) If the point \( M \) does not lie on the segment \( M_{1} M_{2} \) and the vectors \( \overline{M_{1} M_{2}}, \overline{M_{1} M} \) are directed oppositely, then \( k<0 \);

c) If the point \( M \) lies on the segment \( M_{1} M_{2} \), then

\[
0 \leqslant k \leqslant 1
\]

The endpoints of the segment, points \( M_{1} \) and \( M_{2} \), correspond to the values: \( k=0 \) and \( k=1 \).

Task 2. Prove that if the points \( M_{1}\left(x_{1} ; y_{1}\right) \) and \( M_{2}\left(x_{2} ; y_{2}\right) \) belong to the chord of the circle \( x^{2}+y^{2}-r^{2} \leqslant 0 \), then \( x_{1} x_{2}+y_{1} y_{2}-r^{2} \leqslant 0 \).

Hint. Consider the expression \( x_{1} x_{2}+y_{1} y_{2}-r^{2} \) as the algebraic sum of the numerical values of the areas of rectangles with sides \( x_{1} \) and \( x_{2} \), \( y_{1} \) and \( y_{2} \), \( r \) and \( r \) (the numerical values of \( x_{1} x_{2} \) and \( y_{1} y_{2} \) can be positive, negative, or zero).