Circle Geometry: Distance, Equation, Ratios, and Segments

Formula for the distance between two points

The formula for the distance between two points $P_1 (x_1,y_1 )$ and $P_2 (x_2,y_2 )$ in the Cartesian plane is given by the distance formula:
$d=\sqrt{(x_2-x_1 )^2 + (y_2-y_1 )^2 }$, where $d$ is the distance between the two points.

The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In the Cartesian plane, the distance between two points can be thought of as the hypotenuse of a right triangle, with the $x$- and $y$-differences between the two points as the other two sides.
To derive the distance formula, we can draw a right triangle with one side along the $x$-axis and the other side along the $y$-axis. The length of the hypotenuse is the distance between the two points. The $x$-difference between the two points is $x_2 - x_1$, and the y-difference is $y_2 - y_1$. Applying the Pythagorean theorem, we have: $$\text{(distance)}^2= \text{(length of x-difference)}^2 + \text{(length of y-difference)}^2 \rightarrow d^2= (x_2 - x_1 )^2 + (y_2 - y_1 )^2$$

Taking the square root of both sides gives us the distance formula: $$d=\sqrt{(x_2-x_1 )^2 + (y_2-y_1 )^2 }$$ The distance formula can be used to find the distance between any two points in the Cartesian plane. It is a fundamental formula in geometry and is used in various applications, such as finding the shortest distance between two points, calculating the distance traveled by an object, and solving optimization problems.

Equation of a circle

The equation of a circle in the Cartesian plane is given by: $$(x-h)^2 +(y-k)^2 =r^2$$ where $(h,k)$ is the center of the circle and $r$ is the radius.
The equation of a circle is derived from the distance formula, which states that the distance between two points $(x_1,y_1 )$ and $(x_2,y_2 )$ is given by: $$d=\sqrt{(x_2-x_1 )^2 +(y_2-y_1 )^2}$$ In the case of a circle, every point on the circumference of the circle is equidistant from the center. Therefore, if we have a point $(x,y)$ on the circumference of the circle with center $(h,k)$ and radius $r$, we can set $d = r$ in the distance formula: $$r= \sqrt{(x-h)^2+(y-k)^2}$$ Squaring both sides of this equation and simplifying gives us the equation of the circle: $$(x-h)^2+(y-k)^2=r^2$$ This equation shows that the set of all points $(x,y)$ that satisfy the equation is the circumference of a circle with center $(h,k)$ and radius $r$.

Coordinates of points on the circle and trigonometric ratios

When dealing with circles, it is often useful to know the coordinates of points on the circle and the trigonometric ratios associated with these points. Let's consider a circle of radius r centered at the origin $(0,0)$ with a point $P(x,y)$ on the circle.
Here are some important concepts related to the coordinates of points on the circle and trigonometric ratios:

Coordinates of points on the circle: If a point $P(x,y)$ lies on the circle of radius $r$, then we have $$x^2 + y^2 = r^2$$ This means that if we know the value of $r$ and the coordinates of one point on the circle, we can find the coordinates of any other point on the circle.
To find the coordinates of points on a circle, we first need to know the center of the circle and its radius. Once we have this information, we can use trigonometry to find the coordinates of any point on the circle.
Suppose the center of the circle is at the point $(h, k)$ and the radius is $r$. Let $\theta$ be the angle between the positive $x$-axis and the line connecting the center of the circle to the point of interest on the circle (measured counterclockwise). Then the coordinates of the point on the circle are: $$x = h + rcos( \theta )$$ $$y = k + rsin( \theta )$$ Here, $cos( \theta )$ and $sin( \theta )$ are the cosine and sine functions of the angle $\theta$, respectively.

Trigonometric ratios: Given a point $P(x,y)$ on the circle, we can define six trigonometric ratios based on the angle $\theta$ formed between the line passing through the origin and the point $P$, and the positive $x$-axis. These ratios are:

• Sinus: $\sin \theta = \frac{y}{r}$
• Kosinus: $\cos \theta = \frac{x}{r}$
• Tangens: $\tan \theta = \frac{y}{x}$
• Kosekans: $\csc \theta = \frac{r}{y}$
• Sekans: $\sec \theta = \frac{r}{x}$
• Kotangens: $\cot \theta = \frac{x}{y}$

Note that these ratios depend only on the angle $\theta$ and the radius $r$ and not on the coordinates of the point $P$.

Trigonometric functions of special angles: For certain angles, the trigonometric ratios have special values. For example, if $\theta = 0$, then $P$ lies on the positive $x$-axis, so $sin \theta = 0$, $cos \theta = 0$ and $tan \theta = 0$.
Similarly, if $\theta = \frac{\pi}{2}$, then $P$ lies on the positive $y$-axis, so $sin \theta = 1$, $cos \theta = 0$ and $tan \theta$ is undefined.

Applications: The coordinates of points on the circle and trigonometric ratios are used in various applications, such as in navigation and engineering. For example, in surveying, the distance between two points can be calculated using trigonometry, and in engineering, trigonometry is used to calculate the angles of support beams and the lengths of cables.

In summary, the coordinates of points on the circle and trigonometric ratios are important concepts when dealing with circles. The coordinates of a point on the circle can be found given the radius and the coordinates of another point, while the trigonometric ratios are based on the angle formed between the line passing through the origin and a point on the circle. These ratios have special values for certain angles and are used in various applications such as navigation and engineering.

Sector and segment of the circle

The circle is one of the most important geometric figures, and it has several parts that are of interest, including the sector and the segment.

Sector of a Circle: A sector of a circle is the region bounded by two radii and an arc. The angle formed by the two radii is called the central angle of the sector. A sector of a circle is like a slice of a pizza.
The area of a sector of a circle can be found using the following formula: $$A = \frac{\theta}{360^circ} \cdot \pi r^2$$ where $r$ is the radius of the circle and $\theta$ is the central angle of the sector in degrees.
Note that in the formula, the angle $\theta$ must be in degrees. If $\theta$ is given in radians, it must be converted to degrees first by multiplying by $\frac{180}{\pi}$ before using the formula.
To find the area of a sector of a circle, we use the formula: $$A= \frac{1}{2} r^2 \theta$$ where $r$ is the radius of the circle and $\theta$ is the central angle of the sector in radians.

Segment of a Circle: A segment of a circle is a part of the circle that is bounded by a chord and the arc that it cuts off. There are two types of segments: major segment and minor segment. The major segment is the part of the circle that is outside the chord, while the minor segment is the part of the circle that is inside the chord.
To find the area of a segment of a circle, we use the formula: $$A = \frac{1}{2} r^2 (\theta -sin \theta)$$ where $r$ is the radius of the circle and $\theta$ is the central angle of the segment in radians.

The length of the arc of a sector is given by the formula: $L=r \theta$, where $L$ is the length of the arc and $\theta$ is the central angle of the sector in radians.
These formulas can be used in a variety of geometry problems involving circles, such as finding the area of a pizza slice or the area of a section of a circular garden. They can also be used in calculus to find the derivatives of functions involving circles, such as the derivative of the area of a sector with respect to the central angle.