Example 1. Segment $AB$ is divided by point $C(-3,0)$ in the ratio $\lambda=\frac{AC}{CB}=\frac{2}{3}$. Find the length of $AB$, if point $A(-5,-4)$ is given.
Solution. 1) To find the required length using the formula from point $2^{\circ}$, it is necessary to know the coordinates of point $B\left(x_{2}, y_{2}\right)$, which we will determine using the formulas from point $3^{\circ}$.

2) We have:

$$
-3=\frac{-5+\frac{2}{3} x_{2}}{1+\frac{2}{3}}, \quad 0=\frac{-4+\frac{2}{3} y_{2}}{1+\frac{2}{3}}
$$

from which $x_{2}=0, y_{2}=6$. Thus, $B(0,6)$.
3) $A B=\sqrt{(0+5)^{2}+(6+4)^{2}}=\sqrt{125}=5 \sqrt{5}$.

Answer. $5 \sqrt{5}$

## Exercises

1. Given the coordinates of the vertices of triangle $A B C: A(2,-5)$, $B(5,-1), C(-4,3)$
1) Find the lengths of the sides and determine the type of triangle by its angles.
2) Find the lengths of the median $B M$, the altitude $C H$, and the angle bisector $A D$.
3) Find the area of the triangle and the coordinates of points $M$ and $D$. Make a sketch.
2. Given triangle $A B C$ with vertices $A(5,5), B(1,-2), C(-1,2)$.
1) Determine the type of triangle.
2) Find the coordinates of the center of the circle circumscribed around the triangle.
3) Find the area of the triangle and angle $C$. Make a sketch.
3. Given three consecutive vertices of a parallelogram $A(11,4)$, $B(-1,1), C(5,7)$. Determine the coordinates of the fourth vertex.
4. Given the vertices of a triangle $A\left(x_{1}, y_{1}\right), B\left(x_{1}, y_{2}\right), C\left(x_{3}, y_{3}\right)$. Determine the coordinates of the point of intersection of the medians of the triangle.
5. Segment $A D$ is divided into three equal parts by points $B(0,-1)$, $C(2,-3)$. Find the coordinates of the endpoints of segment $A D$.
6. A segment between points $A(0,2)$ and $B(8,0)$ is divided in the same ratio as the distances of these points from the origin. Find the coordinates of the division point.
7. A segment is drawn from point $A(1,-1)$ to point $B(-4,5)$. To what point should it be extended in the same direction to double its length?

## Answers

1. 2) $5,10, \sqrt{97}$, acute-angled; 2) $\left.6, \frac{16}{3}, \frac{48}{5} ; 3\right) S=24, M(-1,1)$, $D\left(2, \frac{1}{3}\right)$. 2. 1) right-angled; 2) $\left.O(2,5,1,5) ; 3\right) 15,90^{\circ}$. 3. $D(17,12)$. 4. $\left(\frac{1}{3}\left(x_{1}+x_{2}+x_{3}\right), \frac{1}{3}\left(y_{1}+y_{2}+y_{3}\right)\right)$. 5. $(-2,1),(4,-5)$. 6. $(1,6,1,6)$. 7. $(-9,11)$.

## § 2. Polar Coordinates

$1^{\circ}$. If a rectangular coordinate system is defined by two mutually perpendicular coordinate axes $O x$ and $O y$, then a polar coordinate system is defined by a single ray (for example, $O x$), which is denoted by $O r$ and called the polar axis, and the point $O$ is the origin of the polar axis, or the pole. In a polar coordinate system, the position of a point $M$ on the plane is determined by two numbers: the angle $\varphi$ (in degrees or radians) that the ray $O M$ makes with the polar axis, and the distance $r=O M$ of the point $M$ from the origin of the polar axis. We write $M=M(r, \varphi)$. For the point $O: r=0, \varphi$ - any.

If the rotation from the axis $O r$ to $O M$ is counterclockwise, then $\varphi$ is considered positive (Fig. 2.3, a), otherwise it is negative.
![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-021.jpg?height=304&width=914&top_left_y=1154&top_left_x=183)

Fig. 2.3

The variable ray $O M$ describes the entire plane if $\varphi$ varies within the range $[0 ; 2 \pi)$ or $(-\pi ; \pi]$.

Sometimes it makes sense to consider $\varphi \in(-\infty ;+\infty)$. In this case, the ray $O M$ describes the plane an infinite number of times (sometimes it is said that $O M$ describes an infinite number of planes).

$2^{\circ}$. The axis $O r$ can be aligned with the ray $O x$ of the rectangular system $O x y$ to establish a connection between the polar coordinates of point $M$ and the rectangular coordinates (Fig. 2.3, b). We have the obvious formulas:

$$
x=r \cos \varphi, \quad y=r \sin \varphi
$$

Formulas (1) express rectangular coordinates in terms of polar coordinates.
Polar coordinates are expressed in terms of rectangular coordinates by the formulas

$$
r=\sqrt{x^{2}+y^{2}}, \quad \operatorname{tg} \varphi=\frac{y}{x} \quad(x \neq 0) .
$$

The formula $\operatorname{tg} \varphi=\frac{y}{x}$ determines two values of the polar angle $\varphi$. Of these two values, the one that satisfies equations (1) should be taken.

$3^{\circ}$. If in the system $O x y$ it is customary to deal with the function $y=y(x)$ (although $x=x(y)$ is also possible), then in the polar system $O \varphi r$ it is equally customary to deal with the function $r=r(\varphi)$.

$4^{\circ}$. The construction of the curve $r=r(\varphi)$ is performed by points (the more points, the better) with due account of the proper analysis of the function $r=r(\varphi)$, justified conclusions about its properties, and the accuracy of the visual estimation when drawing the line.

## Examples with Solutions