영어로 배우는 중학교 수학 - 원에 대하여 / CIRCLES ( 원주, 호, 반지름 등등 ) [ 영문 ]

[CLARK]

  영어로 배우는 중학교 수학    

   원에 대하여  - Circles  ( 원주, 호, 반지름 등등 )    [ 영문 ]  

---

     Video  Lectures    

Watch Video on Area of a Circle - Math Help

Watch Video on Area and Circumference of Circle

     

Watch Video on Area and Circumference of a Circle - Geometry Help
 ------------------------------------------------------------------

  ○ Circles        

  Diameter   

The diameter of a circle is a line segment that passes through the center and has its endpoints on the circle. All diameters of the same circle have equal lengths.

    Radius  

The radius of a circle is a line segment extending from the center of the circle to a point on the circle. In the figure shown below,  OB and OA are radii.

All radii of the same circle have equal lengths, and the radius is half the diameter.

In the figure, OB = OA.

    Arc     

An arc is a part of a circle.  In the figure above, the points on the circle from A to B form an arc.  An arc can be measured in degrees or in units of length. 

    If you form an angle by drawing radii from the ends of the arc to the center of the circle, the number of degrees in the arc (arc AB in the figure ) equals the number of degrees in the angle formed by the two radii at the center of the circle (∠AOB), called the central angle.

Tangent  to  a  Circle    

A tangent to a circle is a line that intersects the circle at exactly one point.  In the figure, line AC is a tangent.  A tangent to a circle is always perpendicular to the radius that contains the one point of the line that touches the circle.  In this case, OA ⊥ AC.

 Circumference    

The circumference is the distance around a circle, and it is equal to π times the diameter, d ( or times twice the radius, r ) .

                                            Circumference = πd        
                                            Circumference = 2 πr            

If the diameter is 16, the circumference is 16π.  If the radius is 3, the circumference is 2(3)π , or 6π   .                          

   Area      

The area of a circle is equal to π  times the square of the radius.

                                           Area =  πr²        
                   

   Example            

In the figure shown below, A is the center of a circle whose area is 25π. B and C are points on the circle. The measure of angle ACB is 45°.  What is the length of line segment  BC? 

How to solve :      

*  Point A is the center of the circle.

 * That makes both  AB and  AC radii, which means that they have equal length.

 * Because AB = AC, △ABC is an isosceles triangle.  The angle opposite AB has a measure of 45°.

 * That means the angle opposite the other equal side is also 45°.

 * The remaining angle is 90°.

* The area of the circle is 25π .

* The formula for the area of a circle is πr².  You can use that formula to figure out the length of the radius, r.

 * That length, r, is also the length of the legs of the triangle whose hypotenuse (BC) has the length you are trying to figure out.

What is the value of r?

                                                       Area   =  πr²      

                                                         25π  =  πr²    

                                                             25 = r²             

                                                               5 = r          

Figuring out the final answer to the problem is a simple matter of working through the

Phythagorean Theorem or remembering that the ratio of the sides of 45° - 45° - 90°

Triangles is 1 : 1 : Root 2 .  The answer is 5 Root 2