In the earlier classes, we have learnt about circles and the terms related to the circles.
Now let us learn about the tangent to a circle.
Secant:
A line which intersects a circle at two distinct points is called a secant of the circle.
If a line \(AB\) cuts a circle at points \(P\) and \(Q\), then that line is called a secant.
Chord:
The line segment inscribed in a circle is called the chord of the circle.
A chord is a sub-section of a secant.
Tangent:
If a line touches the given circle at only one point, then it is called 'tangent' to the circle.
For example,
When a bullock cart moves along a road, then the road becomes the tangent at each point when the wheels roll on it.
Important!
The common point where the circle and the tangent intersect is called the point of contact.
Theorem on circles and tangents:
Statement:
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof:
Consider a circle with centre \(O\).
Let \(AB\) be the tangent to the circle at the point \(P\).
To prove:
The line \(OP\) is perpendicular to \(AB\).
Proof:
Take a point \(Q\) other than \(P\) on the tangent \(AB\) and join \(OQ\).
Here, \(Q\) must lie outside the circle.
Thus, \(OQ\) is longer than \(OP\).
That is \(OQ>OP\) at every point on \(AB\) except at \(P\).
Therefore, the point \(P\) is at the shortest distance from the centre \(O\).
We know that:
Out of all the line segments drawn from the point to points of a line not passing through the point, the smallest is the perpendicular to the line.
By the theorem, \(OP\) is perpendicular to \(AB\).
Hence, the proof.
Example:
In the above given figure, if \(OP = 3\ cm\) and \(PQ = 4\ cm\), find the length of \(OQ\).
Solution:
By the result, \(\angle OPQ = 90^\circ\)
So, \(OPQ\) is a right-angled triangle.
By the Pythagoras theorem, we have:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
\(OQ^2 = OP^2 + PQ^2\)
\(OQ^2 = 3^2 + 4^2\)
\(OQ^2 = 9+16\)
\(OQ^2 = 25\)
\(OQ = \sqrt{25}\)
\(OQ = 5\)
Therefore, the measure of \(OQ = 5\ cm\).
Important!
- By this theorem, we can say that at any point on the circle, there can be one and only one tangent.
- The line containing the radius through the point of contact is also called the normal to the circle at the point.