Circle - Tangent for class 10th students and grade x maths and mathematics students.

Editor's Notes

• #2: Welcome to “tute”
• #3: Today we are going to study about “Tangent to Circle”
• #4: We already know… Definition of circle Circle basics (chord, segment, sector, arc, secant, etc) Intersecting line, non-intersecting line and tangent.
• #5: In this video we will learn.. *Tangent to a circle is a line that intersects the circle at only one point. Danish mathematician Thomas Fineke There is only one tangent at a point of the circle The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
• #6: Consider a circle and draw a tangent to that circle… We can see that, this tangent touches the circle at point A only. So, we can say that tangent to a circle is a line that intersects the circle at only one point. So if we draw tangents to various points on a circle, we will see that they touches the circle at one and only one point respectively. We know that circle is made by joining infinite number of points which are at a same distance from the center. Now if circle consists of infinite number of points, then how many tangents can be drawn to a circle ? If you are thinking that there can be infinite number of tangents drawn, then you are absolute correct. A circle have infinite number of tangents..
• #7: Lets perform an activity to understand “tangent to a circle” in a better way. Take a circular ring and attach a wire AB at a point P of the circular ring so that it can rotate about the point P in a plane. Put the system on a table and gently rotate the wire AB about the point P to get different positions of the straight wire. In various positions, the wire intersects the circular wire at P and at another point Q1 or Q2 or Q3, etc. In one position, you will see that it will intersect the circle at the point P only. This shows that a tangent exists at the point P of the circle. On rotating further, you can observe that in all other positions of AB, it will intersect the circle at P and at another point, say R1 or R2 or R3, etc. So, you can observe that there is only one tangent at a point of the circle. While doing activity, you must have observed that as the position AB moves towards the position A′B′, the common point, say Q1, of the line AB and the circle gradually comes nearer and nearer to the common point P. Ultimately, it coincides with the point P in the position A′ B′ of A ′′B′′. Again note, what happens if ‘AB’ is rotated rightwards about P? The common point R3 gradually comes nearer and nearer to P and ultimately coincides with P. So, what we see is: The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
• #8: The word “TANGENT” comes from the Latin word ‘tangere’, which means to touch and was introduced by the Danish mathematician Thomas Fineke in 1583.
• #9: Activity 2 : On a paper, draw a circle 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 P′Q′ and P′′Q′′ of the secant. 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 the secant when both of the end points of the corresponding chord coincide. The common point of the tangent and the circle is called the point of contact and the tangent is said to Touch the circle at the common point.
• #10: 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? 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.
• #11: Now we know… Tangent to a circle is a line that intersects the circle at one and only one point Only one tangent can be drawn at a point of the circle The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
• #12: In the next video.. We shall prove a theorem, which states that, The tangent at any point of a circle is perpendicular to the radius through the point of contact.