Planimetry is a branch of geometry that studies the properties of plane figures. These include not only well-known triangles, squares, rectangles, but also straight lines and angles. 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?
To understand what a central angle is, you need to define a circle. A circle is a collection of all points equidistant from a given point (the center of the circle).
It is very important to distinguish it from a circle. It must be remembered that a circle is a closed line, and a circle is a part of a 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 whose sides intersect the circle at two points. The arc that an angle delimits by its intersection points is called the arc on which the given angle rests.
Consider example 1.
In the picture, angle AOB is central, because the vertex of the angle and the center of the circle are one point O. It rests on the arc AB, which does not contain point C.
How does an inscribed angle differ from a central one?
However, besides the central ones, there are also inscribed angles. What is their difference? Just like the central one, the angle inscribed in a circle rests on a certain arc. But its vertex does not coincide with the center of the circle, but lies on it.
Let's take the following example.
Angle ACB is called an angle inscribed in a circle centered at point O. Point C belongs to the circle, that is, lies on it. The angle rests 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 a circle, and be able to calculate its value in degrees.
So, the central angle is equal to the degree measure of the arc it rests on.
In the picture, the angle AOB rests on the arc AB equal to 66°. So the angle AOB is also equal to 66°.
Thus, the central angles based on equal arcs are equal.
In the figure, arc DC is equal to arc AB. So angle AOB is equal to angle DOC.
How to find an inscribed angle
It may seem that the angle inscribed in the circle is equal to the central angle,which relies on the same arc. However, this is a gross mistake. In fact, even just looking at the drawing and comparing these angles with each other, you can see that their degree measures will have different values. So what is the angle inscribed in the circle?
The degree measure of an inscribed angle is one half of the arc it rests on, or half the central angle if they rely on the same arc.
Let's consider an example. Angle ACB is based on an arc equal to 66°.
So the angle DIA=66°: 2=33°
Let's consider some consequences of this theorem.
- Inscribed angles, if they are based on the same arc, chord or equal arcs, are equal.
- If the inscribed angles are based on the same chord, but their vertices lie on opposite sides of it, the sum of the degree measures of such angles is 180°, since in this case both angles are based on arcs, the total degree measure of which is 360 ° (whole circle), 360°: 2=180°
- If the inscribed angle is based on the diameter of the given circle, its degree measure is 90°, since the diameter subtends an 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 of the central one.
Where can problems on this topic be found? 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 are a topic that is widely and in detailstudied in the school curriculum. Tasks devoted to their properties are found in the main state exam (OGE) and the unified state exam (USE). As a rule, to solve these problems, you should find the angles on the circle in degrees.
Angles based on the same arc
This type of problem is perhaps one of the easiest, since to solve it you need to know only two simple properties: if both angles are inscribed and lean on the same chord, they are equal, if one of them is central, then the corresponding inscribed angle is equal to half of it. However, when solving them, one must 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.
Problem 1
Given a circle centered at point O. Angle AOB is 54°. Find the degree measure of the angle DIA.
This task is solved in one step. The only thing you need in order to find the answer to it quickly is to notice that the arc on which both corners rest is a common one. Seeing this, you can apply the already familiar property. Angle ACB is half the angle AOB. So
1) AOB=54°: 2=27°.
Answer: 54°.
Angles based on different arcs of the same circle
Sometimes the size of the arc on which the required angle rests is not directly specified in the conditions of the problem. In order to calculate it, you need to analyze the magnitude of these angles and compare them with the known properties of the circle.
Problem 2
In a circle centered at O, angle AOCis 120°, and the angle AOB is 30°. Find the corner YOU.
To begin with, it is worth saying that it is possible to solve this problem using the properties of isosceles triangles, but this will require more mathematical operations. Therefore, here we will analyze the solution using the properties of central and inscribed angles in a circle.
So, the angle AOC rests on the arc AC and is central, which means that the arc AC is equal to the angle AOC.
AC=120°
In the same way, the angle AOB rests on the arc AB.
AB=30°.
Knowing this and the degree measure of the entire circle (360°), you can easily find the magnitude of the arc BC.
BC=360° - AC - AB
BC=360° - 120° - 30°=210°
The vertex of the angle CAB, point A, lies on the circle. Hence, the angle CAB is inscribed and equal to half of the arc CB.
CAB angle=210°: 2=110°
Answer: 110°
Problems based on arc ratios
Some problems do not contain data on angles at all, so they need to be searched based only on known theorems and properties of a circle.
Problem 1
Find the angle inscribed in a circle that is supported by a chord equal to the radius of the given circle.
If you mentally draw lines connecting the ends of the segment with the center of the circle, you get a triangle. Having examined it, you can see that these lines are the radii of the circle, which means that all sides of the triangle are equal. We know that all angles of an equilateral triangleare equal to 60°. Hence, the arc AB containing the vertex of the triangle is equal to 60°. From here we find the arc AB, on which the desired angle is based.
AB=360° - 60°=300°
Angle ABC=300°: 2=150°
Answer: 150°
Problem 2
In a circle centered at point O, the arcs are related as 3:7. Find the smaller inscribed angle.
For the solution, let's denote one part as X, then one arc is equal to 3X, and the second, respectively, 7X. Knowing that the degree measure of a circle is 360°, we can write an equation.
3X + 7X=360°
10X=360°
X=36°
According to the 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 required (smaller) angle corresponds to an arc equal to 3X.
So the smaller angle is (36°3): 2=108°: 2=54°
Answer: 54°
Problem 3
In a circle centered at O, the angle AOB 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 the smaller arc relates to the larger one. To do this, we calculate the magnitude of both arcs in degrees. The smaller arc is equal to the angle that rests on it. Its degree measure is 60°. The larger arc is equal to the difference between the degree measure of the circle (it is equal to 360° regardless of other data) and the smaller arc.
The big arc is 360° - 60°=300°.
Since 300°: 60°=5, the larger arc is 5 times the smaller one.
Big arc=505=250
Answer: 250
So, of course, there are othersapproaches to solving similar 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, you need to carefully study the drawing and compare it with the data of the problem, as well as be able to apply your theoretical knowledge in practice.
