Working Model of Class 9th (Circle Properties) | Explanation | Project File

^{(Circle Properties)}

^{WE SUGGEST THE BEST WAY TO EXPLAIN THE TOPICS OF MATHS}

^{·       Introduction: - By using this working model you can easily explain the various terms of the circle like a chord, secant, sector, and some circle theorems.}

Firstly consider this circle; we can draw a line in three different ways. 

(I) The line PQ and the circle have no common point. In this case, PQ is called a non-intersecting line with respect to the circle. 

(ii) There are two common points A and B that the line PQ and the circle have. In this case, we call the line PQ a secant of the circle.

 

(iii) There is only one point A which is common to the line PQ and the circle. In this case, the line is called a tangent to the circle

How we can prove by using this working model?

On a Circular Cardboard, and a secant PQ of the circle Draw various lines parallel to the secant on both sides of it. You will find that after some steps, the length of the chord cut by the lines will gradually decrease, i.e., the two points of intersection of the line and the circle are coming closer and closer. In one case, it becomes zero on one side of the secant, and in another case, it becomes zero on the other side of the secant. See the positions XY and R  in Fig. 
These are the tangents to the circle parallel to the given secant PQ. This also helps you to see that there cannot be more than two tangents parallel to a given secant. This activity also establishes, what you must have observed, while doing Activity 1, namely, a tangent is a secant when both of the endpoints of the corresponding chord coincide. The common point of the tangent and the circle is called the point of contact [the point A in Fig 

Theorem of Circle

Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

I.            The tangent is said to touch the circle at the common point. Now, look around you. Have you seen a bicycle or a cart moving? Look at its wheels. All the spokes of a wheel are along its radii. Now note the position of the wheel with respect to its movement on the ground. Do you see any tangent anywhere?

II.             In fact, the wheel moves along a line which is a tangent to the circle representing the wheel. Also, notice that in all positions, the radius through the point of contact with the ground appears to be at right angles to the tangent we shall now prove this property of the tangent.

Prove of this theorem in NCERT

Remarks: 1. by the theorem above, we can also conclude that at any point in a circle there can be one and only one tangent.

 2. The line containing the radius through the point of contact is also sometimes called the ‘normal’ to the circle at the point.