The circle, the circumference and their elements. The central angle презентация

You will be able to:
know the definitions of a circumference and a circle, and their elements (center, radius, diameter, chord);
know and apply definition and properties of central angle;
prove and apply the perpendicular diameter-chord theorem;
know the definition of a locus.
CHAPTER no 4
The circle, the circumference and their elements. The central angle.

Key Terms:
circle – шеңбер - окружность
center - центр
radius - радиус diameter - диаметр
chord - хорда
central angle – центрлік бұрыш – цетральный угол
locus – нүктелердің геометриялық орны – геометрическое место тосек
CHAPTER no 4
The circle, the circumference and their elements. The central angle.

You can see many circular or ring-shaped geometric figures all around you.
CHAPTER no 4
The circle, the circumference and their elements. The central angle.

It is easy to recognize a circle, but how can we define it as a shape? Let us look at a geometric definition.
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The word ‘circle’ is derived from the latin word circus, which means ‘ring’ or ‘racecourse’.
Note:
The circle, the circumference and their elements. The central angle.

Definition:
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A CIRCLE is the set of all the points in a plane that are at the same distance from a fixed point in the plane and the fixed point
is called the center of the circle.
The circle, the circumference and their elements. The central angle.

Definition:
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The RADIUS is the distance from the center
The DIAMETER goes straight across the circle, through the center
The CHORD is a line segment which
joins two different points on a circle.
The circle, the circumference and their elements. The central angle.

Definition:
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Circumference is the distance once around the circle
Circumference = π × Diameter
The circle, the circumference and their elements. The central angle.

Example:
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You walk around a circle which has a diameter of 100m, how far have you walked?
Solution:
Distance walked = Circumference = π × 100m
= 314m (to the nearest m)
The circle, the circumference and their elements. The central angle.

Example:
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The length of the diameter of a circle is 20 cm and the radius is 2x – 4. Find x.
Solution:
Diameter = 2 ⋅ Radius
d = 2 ⋅ r
20 = 2 ⋅ (2x – 4)
2x – 4 = 10
2x = 14
x = 7 cm
The circle, the circumference and their elements. The central angle.

CHAPTER no 4
The circle, the circumference and their elements. The central angle.
Definition:
An ARC of a circle consists of two points on the circle and the unbroken part of the circle between these two points.
Any two points of a circle divide the circle into two arcs. If the arcs are unequal, the smaller arc is called the minor arc and the larger arc is called the major arc.
NOTE:

Definition:
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The circle, the circumference and their elements. The central angle.
A CENTRAL ANGLE of the circle is an angle whose vertex is at the center of a circle.
The measure of a minor arc is equal to the measure of its central angle.
NOTE:
Central Angle = Minor Arc
∠APB = A͡B

Examples:
CHAPTER no 4
The circle, the circumference and their elements. The central angle.
Find the measure of the indicated central angle of each circle.

Theorem:
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The circle, the circumference and their elements. The central angle.
If a diameter or radius is perpendicular to a chord, then it bisects the chord and its arc.
In the figure, if [OH] ⊥ [AB] then |AH| = |HB|.
Example:

Example:
CHAPTER no 4
The circle, the circumference and their elements. The central angle.
A chord of length 10 cm is 12 cm away from the center of a circle. Find the length of the radius.
In ΔAHO, r² = 5² + 12²
r² = 25 + 144
r² = 169
r = 13 cm.
Solution:
Look at the figure.

Definition:
CHAPTER no 4
The circle, the circumference and their elements. The central angle.
LOCUS is the set of all points that share a property. This usually results in a curve or surface.
Example:
A Circle is "the locus of points on a plane that are a certain distance from a central point".

Activity:
CHAPTER no 4
The circle, the circumference and their elements. The central angle.
Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!

Resources :
CHAPTER no 4
The circle, the circumference and their elements. The central angle.
https://www.mathsisfun.com/geometry/circle.html https://www.onlinemathlearning.com/diameter-of-circle.html
https://www.mathemania.com/lesson/the-definition-and-basics/
https://www.khanacademy.org/math/basic-geo/basic-geo-area-and-perimeter/area-circumference-circle/v/parts-of-a-circle
https://www.mathplanet.com/education/geometry/circles/basic-information-about-circles
https://www.brainfuse.com/jsp/alc/resource.jsp?s=gre&c=37818&cc=108830