The circle is the set of all points of the plane that are equally distanced from one fixed point of the same plane that we call center of the circle.
The distance from the center to any point on the circle is called the radius of the circle.
Chord is the line that connects two different points that lies on it.
The longest chord is called the diameter. Center lies on the diameter. The length of a diameter is equal to .
Sector is an enclosed part of a circle which contains two radius and a part of a circumference.
Segment is a part of a circle enclosed by a chord and a part of the circumference.
Secant is the line that intersects the circle in two points and tangent is a line that has exactly one point in common with the circle.
Concentric circles have the same center but different radius.
If r is the length of the circle, then its circumference is given with the formula:
and area with
From this we can see that the number is defined as the ration of the length of circumference and its diameter.
Length of the arc and area of a sector
Firstly, let’s observe circle on which lie two different points A and B. Those points divide it in two different arcs. Their union is the whole circle.
For denoting arcs we have to refer to the positive direction – opposite of the direction of the clocks hands.
This means that
We can draw angle whose axis is the center of the circssle and whose legs go through the points A and B. this angle is called the central circle.
If the central angle has degrees; than the length of the arc is α times greater than the arc that matches the angle.
In the similar way we can find the formula for the area of the sector.
Central and inscribed angles
Every angle whose vertex is a point on the circle and whose legs cut the same circle is called inscribed angle.
If we mark the points in which the legs of the inscribed angle cut the circle with points A and B, then we call that angle the inscribed angle over the segment AB.
For these two points we can also define another angle whose legs also go through those points, but its vertex is placed in the center. This is called the central circle.
The central and inscribed angle theorem
The central angle over the arc of the circle is equal to the double inscribed angle over that same arc.
Proof. There are exactly three different cases that may appear.
1. Center of the circle lies on one leg of the inscribed angle
In this case we can observe triangle TSB.
Which means that triangle TSB is isosceles triangle which implies that angles STB and SBT are equal. is the external angle of triangle TSB which means that he is equal to the sum of two opposite angles – STB and SBT.
This leads us to
2. Center of the circle lies within inscribed angle.
For the second case we can divide these two angles in half. By doing so, we simplified the task and we can see that we only have to apply the first case two times.
3. The center lies outside of the inscribed circle
For proof of this case we’ll also use case 1.
When we add these two equations we get
Example 1. Find the missing angle:
Example 2. Find the missing angle.
Every inscribed angle over diameter of the circle is the right angle.
Proof. First, draw a circle and a triangle whose one side is the diameter and whose vertex is any point on the circle different from A and B.
Draw a line that connects C and O.
Triangle is isosceles.
Triangle BOC is isosceles.
Reverse of Thales’ theorem
Center of the circumscribed circle of right triangle is located in the middle of hypotenuse.
Divide the angle at point c with a line cx in a way that .
Triangle ACX is isosceles
Triangle CBX is isosceles!
This means that x is the center of circumscribed circle and middle point of hypotenuse.
Every curve has its equation. From that equation we can find all the information we need to draw that curve and know every one of its properties.
Circles’ equation can be derived from its definition.
Let’s say that the coordinate of the circles center is S(p, q) and any point T has coordinate T(x, y). From the formula for the distance of two points and the fact that we know that that distance is equal to the length of the radius r we get that:
If S(p, q) is the center of the circle, and r its radius, then the equation is
What if we put the center of the circle in origin and set the radius of 1? We get the unit circle.
Its equation is:
Example 1. Draw the circle that is described by the following equation.
From the left side of this equation we can find the center of the circle. The x coordinate of the center is 2, and the y coordinate is – 2.
From the right side of the equation we can find radius. Radius is equal to . Since radius can only be a positive number we can see that the radius is equal to 3.
Mutual position of line and circle
Line and circle can be in three positions.
1. Two different points – line is the secant of the circle.
2. One point in common – line is the tangent of the circle.
3. No points in common
How can you find out in which relation are the circle and the line without having to draw them? You can do that using their equations.
To find their relation we have to find their common points.
Example 1. In which relation are the line 2x – y + 3 = 0 and the circle ?
First, from the line equation we extract y or x. in this example we’ll extract y.
y = 2x + 3
And then insert this y into the equation.
By solving this quadratic equation we get the solutions , . By inserting them into any of these equations we get matching y coordinates and finally the points:
. This means that the given line and circle intersect in two points.
If we got only one point this would mean that the line is tangent of the circle and if the solutions were complex this would mean that the line and the circle have no points in common.
Let k be a circle and T any point of the plane. For any line p which goes through point T, the product is a constant, where A and B are the intersections of the line p and circle k.
1. T lies on the circle.
This means that T = A or T = B which means that either or
2. T lies within the circle.
This means that the triangles ACT and BTD are similar.
3. T lies outside of the circle.
First we’ll draw tangent on the circle from the point T.
Triangle SNA is congruent to triangle SNB – >
This means that this circle power does not depend on A and B.
Real number is called circle power of the point T considering circle k.
Tangent and normal of the circle
The equation of the tangent
with touching point is
Normal is a line that is perpendicular to the tangent and goes through the touching point of the tangent and the circle.
The equation of the normal
With touching point of the tangent and the circle is
Example. Determine the equations of tangent and normal on the circle in the point T with coordinates T (1, 4)
First thing we have to do is convert this equation to a form we are used to.
Now we can clearly see everything we have to know if we want to find equations of normal and the tangent.
Constructing circles (174.0 KiB, 147 hits)
Angles (375.8 KiB, 136 hits)
Arcs (273.7 KiB, 121 hits)
Measurement of angles (455.2 KiB, 229 hits)
Measurement of arcs (425.5 KiB, 142 hits)
Measure the area and circumference (171.1 KiB, 147 hits)
Measure the inscribed angles of a circle (735.9 KiB, 138 hits)
Measure the angle between secants and tangents (710.6 KiB, 142 hits)
Tangents (589.7 KiB, 148 hits)
Arc length (239.4 KiB, 146 hits)
Sector area (239.6 KiB, 146 hits)
Finding different segment measures - easy (424.1 KiB, 145 hits)
Finding different segment measures - advanced (390.5 KiB, 135 hits)
Make equations for the circle (670.4 KiB, 141 hits)