In the previous section we completed a discussion on Coordinate geometry. In this chapter we will learn about Circles and Tangents.
We have learned about circles in a previous chapter 21.
• A circle is a collection of points. All those points are at equal distances from a fixed point.
♦ The 'equal distances' is the radius of the circle.
♦ The fixed point is the centre of the circle.
• We have also learned about chord, segment, sector, arc etc.,
In this chapter we will see the calculations involving circles and line segments together. Consider fig.32.1 below:
• In fig(a), the line AB lies near a circle. But they are not touching each other. So we say that AB is a non intersecting line with respect to the circle.
• In fig(b), AB intersects the circle at 2 points P and Q. That means, there are two common points for the line AB and the circle. In this case, the line AB is called a secant of the circle.
• In fig(c), there is only one common point for the line PQ and the circle. In this case, the line PQ is called a tangent of the circle.
• The word 'tangent' comes from the Latin word 'tangere' which means 'to touch'
• Compare the lengths PQ and PQ1. We can see that PQ1 is smaller than PQ.
• That means, the chord has become smaller.
4. Let us continue the rotation. P will remain the same. But Q1 will change. Let Q2 be the new position. This is shown in fig(c).
• Compare the lengths PQ1 and PQ2. We can see that PQ2 is smaller than PQ1.
• That means, the chord has become still smaller.
5. Let us continue the rotation. P will remain the same. But Q2 will change. Let Q3 be the new position. This is shown in fig(d).
• Compare the lengths PQ2 and PQ3. We can see that PQ3 is smaller than PQ2.
• That means, the chord has become still smaller.
■ So it is clear that, when we rotate line AB, the chord will become smaller and smaller. What will happen to the chord if we continue the rotation?
Ans: The length of the chord will decrease to such a level that, it is no longer a chord, but a mere point. This is shown in fig(e).
• So in the final stage, line AB is not touching the circle at two points. It is touching the circle at only one point. That means line AB has become a tangent of the circle.
6. Let us continue the rotation even after the stage when the chord reduces to a point. That is., we are going to continue the rotation after the stage shown in fig.32.3(e). For that, consider fig.32.4 below:
7. The first fig. is the same 32.3(e) that we saw earlier. From there, we are going to rotate the line AB in the same counter clock-wise direction as indicated by the green arrow.
8. We can see that, even with the smallest rotation, a second point of intersection will be obtained. Let this new point of intersection be Q4. It is shown in fig(f).
• So the point is no longer a ‘point’. It has become a chord.
• Consequently, the tangent in fig(e) is no longer a tangent. it has become a secant again.
• If we continue the rotation, the length of chord will increase as shown in figs. (g) and (h)
■ So there is only one direction for line AB in which the tangent can be obtained at P. For all other directions, it will be secants.
■ There is only one tangent at a point of the circle
■ The tangent is a special case of the secant. In this special case, the two end points of the chord formed by the secant will coincide at a single point.
2. Suppose that we want to draw a tangent of the circle at that point.
3. Then there is a ‘particular direction’ in which we must draw the tangent at that point.
4. Every point on a circle has it's own 'particular direction'.
5. If we do not draw the line in that direction, the tangent will not be obtained. What we get, will be a secant.
• Later in this chapter, we will learn how to obtain this ‘particular direction’
Let us do one more experiment:
Consider fig.32.5 below:
1. In fig(a), a secant line AB is shown in red colour. It intersects the circle at two points P and Q.
2. Two other secant lines are also drawn in the same fig. One on either side of AB.
• The green lines are drawn exactly parallel to AB.
• Altogether, there are three chords. We can see that, both the green chords are smaller than the red chord.
3. In fig(b), two more green secants are drawn parallel to the red secant. This time also, they are drawn on either sides of AB.
• Again we find that the chords made by the new secants are lesser than those in fig(a).
4. So it is clear that, as we move away from the original secant AB, on either sides, the length of chord becomes smaller and smaller.
• At last we will reach a stage in which the chord will no longer be a chord, but a single point.
• That means, the two end points of the chord will coincide at a single point.
• So, at that stage, the secant will become a tangent.
■ From this experiment, we can write:
1. A circle and any one of it's secant is given
2. Suppose we are asked to draw tangents parallel to the given secant
3. We will be able to draw only two tangents parallel to the given secant
1. In fig(a), the wheel rolls over a level road. The road does not have any ups or downs. So it can be indicated with a straight line.
2. The ends of the spokes successively come in contact with the road. Concentrate on any one spoke.
3. Stop the cycle at the instant when the end of that spoke comes in contact with the road.
4. We will find that, the spoke is exactly perpendicular to the road.
■ Now, what is the peculiarity of the road?
Ans: • It is a straight line, and is touching the circle at only one point. So the road is a tangent to the circle.
■ What is the peculiarity of the particular spoke that we considered?
Ans: • It is a radial line. So it passes through the centre of the circle
• The particular spoke that we considered passes through the point of contact also.
• This particular spoke is perpendicular to the road.
• In this experiment we considered a road which is perfectly horizontal. We will get the same results for an inclined road also. However, the road should not have any ups or downs. That is., we must be able to represent the road by a straight line. This is shown in fig(b).
• At the instant when the end of a spoke comes in contact with the inclined road, it will be exactly perpendicular to the road.
We can write the above findings in the form of a theorem. We will write it in a step by step manner:
Theorem 32.1:
1. Consider any tangent of a circle.
2. Mark the point of contact P.
3. Draw a line from this point of contact to the centre O.
4. Then the line OP will be perpendicular to the tangent
1. In fig(a), AB is a tangent at point P. A radial line is drawn from P to O. We have to prove that OP is perpendicular to AB.
2. Mark a point Q any where on the line AB. Join OQ with a dashed line
3. Consider the length of OQ. This length will always be greater than the radius of the circle. This is because, if the length is less, Q will fall inside the circle, and the tangent AB will become a secant as shown in the fig(b)
4. So, if AB is a tangent at P
AND
Q is any point on AB
THEN OQ will be greater than OP, what ever be the position of Q
5. It follows that, OP is the shortest distance between the centre O and line AB
■ If OP is the shortest distance, then obviously, OP will be perpendicular to AB. Thus the theorem is proved.
2. Draw the radius OP
3. Draw a perpendicular to OP at the end point P. This perpendicular is the required tangent
2. We are drawing it at the end point of the radius, which is marked as P.
3. There is only one such perpendicular possible at P. That means, the tangent at any point is unique.
■ In other words, there is only one tangent possible at any given point of a circle.
We will now see a solved example
Solved example 32.1
A tangent AB touches the circle of radius 5 cm at P. Another point Q on the tangent is at a distance of 12 cm from the centre of the circle. Then what is the distance between P and Q?
Solution:
A rough sketch based on the given data is shown in the fig.32.8(a) below:
From the fig(a), it is clear that:
• P is the point of contact
• Q is another point on the tangent AB
• Distance between O and Q is 12 cm
Based on this rough sketch, we can solve the problem:
1. Given that radius = 5 cm. So OP = 5 cm
2. PQ is a tangent at P. So OP will be perpendicular to PQ
3. Thus we have a right angled triangle: ⊿OPQ
Applying Pythagoras theorem we get:
PQ2 = OQ2 - OP2 ⇒ PQ2 = 122 - 52 ⇒ PQ2 = 144 - 25 = 119 ⇒ PQ = √119
Two actual construction problems are given in the form of a video presentation here.
• A unit circle means that, it's radius is 1 unit.
♦ It can be 1 cm, 1 m etc.,
2. A tangent is drawn from a point T onto the circle. The tangent touches the circle at P.
• We want the length of the tangent. That is., we want the length PT.
3. We know that OP will be perpendicular to the tangent. So we can form a right triangle as shown in fig.b
• Let us denote the angle at O as θ
• Now take the tangent of the angle θ. We get:
tan θ = PT⁄OP = PT⁄1 = PT
4. So length of the tangent = tan θ
In the next section, we will see more details about Tangents.
We have learned about circles in a previous chapter 21.
• A circle is a collection of points. All those points are at equal distances from a fixed point.
♦ The 'equal distances' is the radius of the circle.
♦ The fixed point is the centre of the circle.
• We have also learned about chord, segment, sector, arc etc.,
In this chapter we will see the calculations involving circles and line segments together. Consider fig.32.1 below:
 |
| Fig.32.1 |
• In fig(b), AB intersects the circle at 2 points P and Q. That means, there are two common points for the line AB and the circle. In this case, the line AB is called a secant of the circle.
• In fig(c), there is only one common point for the line PQ and the circle. In this case, the line PQ is called a tangent of the circle.
• The word 'tangent' comes from the Latin word 'tangere' which means 'to touch'
Let us see an example for tangents from our day to day life:
• Consider fig.32.2(a) below. It shows a pulley used to lift loads.
• A rope passes through the pulley. There are two vertical segments for the rope. One on either side.
 |
| Fig.32.2 |
• Each of these vertical segments form a tangent of the circle.
• This will be more clear if we extend each vertical side upwards. The upward extension should be along the same line as the vertical segments. This is shown in fig.32.2(b).
Now we will learn more about tangents. Consider fig.32.3 below:
 |
| Fig.32.3 |
1. In fig(a), a line AB touches the circle at two points P and Q. So AB is a secant.
2. Let AB be pivoted at P. That is., the line AB can rotate freely about P. Let us rotate AB in a counter clock-wise direction as indicated by the green arrow.
3. When the rotation takes place, P will remain the same. But Q will change. Let Q1 be the new position. This is shown in fig(b). • Compare the lengths PQ and PQ1. We can see that PQ1 is smaller than PQ.
• That means, the chord has become smaller.
4. Let us continue the rotation. P will remain the same. But Q1 will change. Let Q2 be the new position. This is shown in fig(c).
• Compare the lengths PQ1 and PQ2. We can see that PQ2 is smaller than PQ1.
• That means, the chord has become still smaller.
5. Let us continue the rotation. P will remain the same. But Q2 will change. Let Q3 be the new position. This is shown in fig(d).
• Compare the lengths PQ2 and PQ3. We can see that PQ3 is smaller than PQ2.
• That means, the chord has become still smaller.
■ So it is clear that, when we rotate line AB, the chord will become smaller and smaller. What will happen to the chord if we continue the rotation?
Ans: The length of the chord will decrease to such a level that, it is no longer a chord, but a mere point. This is shown in fig(e).
• So in the final stage, line AB is not touching the circle at two points. It is touching the circle at only one point. That means line AB has become a tangent of the circle.
6. Let us continue the rotation even after the stage when the chord reduces to a point. That is., we are going to continue the rotation after the stage shown in fig.32.3(e). For that, consider fig.32.4 below:
 |
| Fig.32.4 |
8. We can see that, even with the smallest rotation, a second point of intersection will be obtained. Let this new point of intersection be Q4. It is shown in fig(f).
• So the point is no longer a ‘point’. It has become a chord.
• Consequently, the tangent in fig(e) is no longer a tangent. it has become a secant again.
• If we continue the rotation, the length of chord will increase as shown in figs. (g) and (h)
■ So there is only one direction for line AB in which the tangent can be obtained at P. For all other directions, it will be secants.
■ There is only one tangent at a point of the circle
■ The tangent is a special case of the secant. In this special case, the two end points of the chord formed by the secant will coincide at a single point.
From the above experiment, we can write the following:
1. Consider any point P on a circle2. Suppose that we want to draw a tangent of the circle at that point.
3. Then there is a ‘particular direction’ in which we must draw the tangent at that point.
4. Every point on a circle has it's own 'particular direction'.
5. If we do not draw the line in that direction, the tangent will not be obtained. What we get, will be a secant.
• Later in this chapter, we will learn how to obtain this ‘particular direction’
Consider fig.32.5 below:
 |
| Fig.32.5 |
2. Two other secant lines are also drawn in the same fig. One on either side of AB.
• The green lines are drawn exactly parallel to AB.
• Altogether, there are three chords. We can see that, both the green chords are smaller than the red chord.
3. In fig(b), two more green secants are drawn parallel to the red secant. This time also, they are drawn on either sides of AB.
• Again we find that the chords made by the new secants are lesser than those in fig(a).
4. So it is clear that, as we move away from the original secant AB, on either sides, the length of chord becomes smaller and smaller.
• At last we will reach a stage in which the chord will no longer be a chord, but a single point.
• That means, the two end points of the chord will coincide at a single point.
• So, at that stage, the secant will become a tangent.
■ From this experiment, we can write:
1. A circle and any one of it's secant is given
2. Suppose we are asked to draw tangents parallel to the given secant
3. We will be able to draw only two tangents parallel to the given secant
■ The common point of the tangent and the circle is called the point of contact
■ The tangent is said to touch the circle at the common point
• Consider the movement of a bicycle. Look closely at one of it's wheels.
• The wheel have spokes. Spokes have a peculiarity:
♦ That is., all of them radiate from the exact centre of the wheel.
♦ That means, the spokes are radial lines.
• Consider fig.32.6 below:
 |
| Fig.32.6 |
2. The ends of the spokes successively come in contact with the road. Concentrate on any one spoke.
3. Stop the cycle at the instant when the end of that spoke comes in contact with the road.
4. We will find that, the spoke is exactly perpendicular to the road.
■ Now, what is the peculiarity of the road?
Ans: • It is a straight line, and is touching the circle at only one point. So the road is a tangent to the circle.
■ What is the peculiarity of the particular spoke that we considered?
Ans: • It is a radial line. So it passes through the centre of the circle
• The particular spoke that we considered passes through the point of contact also.
• This particular spoke is perpendicular to the road.
• In this experiment we considered a road which is perfectly horizontal. We will get the same results for an inclined road also. However, the road should not have any ups or downs. That is., we must be able to represent the road by a straight line. This is shown in fig(b).
• At the instant when the end of a spoke comes in contact with the inclined road, it will be exactly perpendicular to the road.
Theorem 32.1:
1. Consider any tangent of a circle.
2. Mark the point of contact P.
3. Draw a line from this point of contact to the centre O.
4. Then the line OP will be perpendicular to the tangent
Now we will write the proof for the theorem:
Consider fig.32.7 below: |
| Fig.32.7 |
2. Mark a point Q any where on the line AB. Join OQ with a dashed line
3. Consider the length of OQ. This length will always be greater than the radius of the circle. This is because, if the length is less, Q will fall inside the circle, and the tangent AB will become a secant as shown in the fig(b)
4. So, if AB is a tangent at P
AND
Q is any point on AB
THEN OQ will be greater than OP, what ever be the position of Q
5. It follows that, OP is the shortest distance between the centre O and line AB
■ If OP is the shortest distance, then obviously, OP will be perpendicular to AB. Thus the theorem is proved.
This theorem gives us a method to draw a tangent at any point. Let us write the steps:
1. Mark the required point P on the circle. We want a tangent at P2. Draw the radius OP
3. Draw a perpendicular to OP at the end point P. This perpendicular is the required tangent
From this theorem, we get another conclusion also. Let us analyse:
1. We are drawing the perpendicular to the radius2. We are drawing it at the end point of the radius, which is marked as P.
3. There is only one such perpendicular possible at P. That means, the tangent at any point is unique.
■ In other words, there is only one tangent possible at any given point of a circle.
Solved example 32.1
A tangent AB touches the circle of radius 5 cm at P. Another point Q on the tangent is at a distance of 12 cm from the centre of the circle. Then what is the distance between P and Q?
Solution:
A rough sketch based on the given data is shown in the fig.32.8(a) below:
 |
| Fig.32.8 |
• P is the point of contact
• Q is another point on the tangent AB
• Distance between O and Q is 12 cm
Based on this rough sketch, we can solve the problem:
1. Given that radius = 5 cm. So OP = 5 cm
2. PQ is a tangent at P. So OP will be perpendicular to PQ
3. Thus we have a right angled triangle: ⊿OPQ
Applying Pythagoras theorem we get:
PQ2 = OQ2 - OP2 ⇒ PQ2 = 122 - 52 ⇒ PQ2 = 144 - 25 = 119 ⇒ PQ = √119
Two actual construction problems are given in the form of a video presentation here.
• The name 'tangent' is derived from the Latin word 'tangere'. It means 'to touch'.
• We have seen the word tangent in another chapter on Trigonometry.
♦ There we saw: Sine, Cosine, Tangent, Cosecant, Secant and Cotangent.
• Is there any relation between the two tangents? Let us see:
1. Consider a unit circle shown in the fig. below:
• A unit circle means that, it's radius is 1 unit.
♦ It can be 1 cm, 1 m etc.,
2. A tangent is drawn from a point T onto the circle. The tangent touches the circle at P.
• We want the length of the tangent. That is., we want the length PT.
3. We know that OP will be perpendicular to the tangent. So we can form a right triangle as shown in fig.b
• Let us denote the angle at O as θ
• Now take the tangent of the angle θ. We get:
tan θ = PT⁄OP = PT⁄1 = PT
4. So length of the tangent = tan θ
In the next section, we will see more details about Tangents.
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