Geometry for grade 8.

Circle:

A circle is a set of point such that every point is at equal distance from a fixed point called the center of the circle, and the set of points are the boundary of the circle. This circle is named by its center .

 

Cord:

Any line segment which has its end points on the boundary of the circle is called the cord.

       

Diameter:

It is the cord passing through the center of the circle.

 

Radius:

It is half of the diameter.

 

Circumference:

It the length of the boundary of the circle is called the circumference of the circle.

 

PI:

It is a special number and it is a ratio between the circumference and diameter of the circle. It is represented by a special symbol “л ”  and its value is 22/7.

 

Circumference of a circle is = 2 л r, where r is the radius of that circle.

 

Area of a circle is the region covered by the circle. And is given as:

Area = л r ².

 

Example:

Find the area and circumference of a circle having radius = 14cm?

Circumference = 2 × л  ×r = 2(22/7)14 = 88cm

Area = л ×r×r= (22/7)(14)(14)= 616cm.square.

  

Some basic facts about geometry:

We need few facts to provide proof for the statements in geometry.

For this some of the results are proved. These proved  results are called axioms and the statements we prove using these axioms are called theorem.

 

Basic Axioms:

1. Infinite lines are passed through one point.
2. We can draw one and only one line with two points.
3. With one center and one given length of radius only one circle can be drawn.
4. A line can be produced to an infinite length from its finite end point.

Now using these axioms we can prove the following theorems. All of these are proved using simple mathematics axioms.

 

Theorems:

1. If we are given two lines of unequal length, then we can divide the greater line in two parts. In which one part has the length equal to the smaller length.
2. Two triangles are said to be congruent, if its two side and one angle are equal to the respective two sides and the angles.
3. Two triangles are said to be congruent if their two angles and one side are congruent to the respective angles and side.
4. For any given straight line of a defined length, we can make an equilateral triangle.
5. Two triangles are said to be congruent if their respective three sides are equal.
6. If two angles of one triangle are equal to the other triangle, then the side on which these triangles are subtended are also equal.
7. A finite straight line can always be bisected.
8. For any isosceles triangle the base angles are equal to one another.
9. We can draw a perpendicular straight line on any line from a point away from it.
10. When a straight line cuts another straight line, the adjacent angles thus formed are always complementary (i.e., their sum is always 90).
11. When two lines cut one another vertical angles thus formed are always equal.
12. The greater angle of a triangle subtends the greater angle.
13. Sum of the two sides of a triangle is always greater than the third side
    

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