Planimetry is a branch of geometry that studies the properties of planar shapes. These include not only well-known triangles, squares, rectangles, but also straight lines and corners. In planimetry, there are also such concepts as angles in a circle: central and inscribed. But what do they mean?
What is the central angle?
In order to understand what a central angle is, you need to give a definition of a circle. A circle is a collection of all points equidistant from a given point (center of a circle).
It is very important to distinguish it from the circle. It must be remembered that the circle is a closed line, and the circle is a part of the plane bounded by it. A polygon or an angle can be inscribed in a circle.
A central angle is an angle whose vertex coincides with the center of the circle, and the sides intersect the circle at two points. The arc that the angle limits to the points of intersection is called the arc on which the given angle rests.
Consider example number 1.
In the picture, the angle AOB is central, because the vertex of the angle and the center of the circle are one point O. It relies on an arc AB that does not contain C.
How does the inscribed angle differ from the central one?
However, besides the central ones, there are also inscribed angles. What is their difference? As well as the central angle inscribed in a circle, it is supported by a certain arc. But its top does not coincide with the center of the circle, but lies on it.
We give the following example.
The angle ACB is called the angle inscribed in a circle centered at point O. Point C belongs to a circle, that is, lies on it. The angle is based on the arc AB.
What is the central angle
In order to successfully cope with problems in geometry, it is not enough to be able to distinguish between inscribed and central angles. As a rule, to solve them, you need to know exactly how to find the central angle in the circle, and be able to calculate its value in degrees.
So, the central angle is equal to the degree measure of the arc on which it rests.
In the picture, the angle AOW is based on the arc AB equal to 66 °. Hence, the angle of AOW is also equal to 66 °.
Thus, the central angles based on equal arcs are equal.
In the figure, the arc DC is equal to the arc AB. So, the angle of AOB is equal to the angle of DOC.
How to find the inscribed angle
It may seem that the angle inscribed in the circle is equal to the central angle, which rests on the same arc. However, this is a gross mistake. In fact, even just by looking at the drawing and comparing these angles with each other, you can see that their degree measures will have different meanings. So what is the angle inscribed in the circle?
The degree measure of the inscribed angle is equal to one second of the arc on which it rests, or half of the central angle, if they rest on one arc.
Consider an example. The ASV angle is based on an arc of 66 °.
Hence, the angle ASV = 66 °: 2 = 33 °
Let us consider some consequences of this theorem.
- Inscribed angles, if they rely on the same arc, chord or equal arcs, are equal.
- If the inscribed angles are based on one chord, but their vertices lie on opposite sides of it, the sum of degree measures of such angles is 180 °, since in this case both angles are based on arcs, the degree measure of which is 360 ° in total (whole circle) , 360 °: 2 = 180 °
- If the inscribed angle is based on the diameter of a given circle, its degree measure is 90 °, since the diameter tightens the arc equal to 180 °, 180 °: 2 = 90 °
- If the central and inscribed angles in a circle are based on the same arc or chord, then the inscribed angle is equal to half the central.
Where problems on this subject can meet? Their types and solutions
Since the circle and its properties are one of the most important sections of geometry, planimetry in particular, the inscribed and central angles in the circle is a topic that is widely and thoroughly studied in the school course. Problems dedicated to their properties are found in the main state exam (USE) and the unified state exam (USE). As a rule, to solve these problems, it is necessary to find the angles on the circle in degrees.
Single Arc Angles
This type of problem is perhaps one of the easiest, because to solve it you need to know only two simple properties: if both angles are inscribed and rely on the same chord, they are equal, if one of them is central, then the corresponding inscribed angle is its half. However, when solving them, you need to be extremely careful: sometimes it is difficult to notice this property, and students when solving such simple problems come to a standstill. Consider an example.
Task number 1
A circle with a center at point O is given. The angle AOB is 54 °. Find the degree measure of the angle of the DIA.
This problem is solved in one action. The only thing needed to find an answer to it quickly is to notice that the arc on which both angles are based is common. Seeing this, you can apply a familiar property. The angle ACB is equal to half the angle AOB. Means
1) AOB = 54 °: 2 = 27 °.
Answer: 54 °.
Angles based on different arcs of the same circle
Sometimes, under the conditions of the problem, the magnitude of the arc on which the desired angle is based is not directly spelled out. In order to calculate it, you need to analyze the magnitude of these angles and compare them with the known properties of the circle.
Task 2
In a circle centered at point O, the AOC angle is 120 °, and the AOW angle is 30 °. Find the angle YOU.
To begin with, it is worth saying that it is possible to solve this problem using the properties of isosceles triangles, however, this will require more mathematical operations. Therefore, here we will analyze the solution using the properties of the central and inscribed angles in a circle.
So, the AOS angle is based on the AC arc and is central, which means that the AC arc is equal to the AOS angle.
AC = 120 °
In the same way, the angle AOW is based on the arc AB.
AB = 30 °.
Knowing this and the degree measure of the entire circumference (360 °), you can easily find the value of the arc of the aircraft.
BC = 360 ° - AC - AB
BC = 360 ° - 120 ° - 30 ° = 210 °
The vertex of the angle CAB, point A, lies on a circle. Hence, the angle of the SAW is inscribed and equal to half the arc of CB.
CAB angle = 210 °: 2 = 110 °
Answer: 110 °
Arc Relationship Tasks
Some tasks do not contain data on the values of the angles at all, so they need to be searched on the basis of only known theorems and properties of the circle.
Task 1
Find the angle inscribed in a circle that rests on a chord equal to the radius of a given circle.
If you mentally draw the lines connecting the ends of the segment with the center of the circle, you get a triangle. Having examined it, you can notice that these lines are the radii of a circle, which means that all sides of the triangle are equal. It is known that all angles of an equilateral triangle are 60 °. Therefore, the arc AB containing the vertex of the triangle is 60 °. From here we find the arc AB, on which the desired angle rests.
AB = 360 ° - 60 ° = 300 °
Angle ABC = 300 °: 2 = 150 °
Answer: 150 °
Task 2
In a circle centered at point O, the arcs are correlated as 3: 7. Find a smaller inscribed angle.
To solve, we denote one part by X, then one arc is 3X, and the second is 7X, respectively. Knowing that the degree measure of the circle is 360 °, we make the equation.
3X + 7X = 360 °
10X = 360 °
X = 36 °
By condition, you need to find a smaller angle. Obviously, if the value of the angle is directly proportional to the arc on which it rests, then the desired (smaller) angle corresponds to an arc equal to 3X.
So the smaller angle is (36 ° * 3): 2 = 108 °: 2 = 54 °
Answer: 54 °
Task 3
In a circle centered at point O, the angle of AOW is 60 °, and the length of the smaller arc is 50. Calculate the length of the larger arc.
In order to calculate the length of a larger arc, you need to make a proportion - how a smaller arc relates to a larger one. To do this, calculate the magnitude of both arcs in degrees. A smaller arc is equal to the angle that rests on it. Her degree measure will be 60 °. The large arc is equal to the difference in the degree measure of the circle (it is 360 ° regardless of the rest of the data) and the smaller arc.
The large arc is 360 ° - 60 ° = 300 °.
Since 300 °: 60 ° = 5, the large arc is 5 times larger than the smaller one.
Large arc = 50 * 5 = 250
Answer: 250
So, of course, there are other approaches to solving such problems, but all of them are somehow based on the properties of central and inscribed angles, triangles and circles. In order to successfully solve them, it is necessary to carefully study the drawing and compare it with the data of the problem, as well as be able to apply their theoretical knowledge in practice.