Circle
Mathematics Class Tenth
circle, tangent & theorem related to tangent & circle
Circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre).
Different Situations when a circle and a line are given in a plane
When a circle and line say PQ are given in a plane, only three possibilities can be occurred.
(a) The line PQ and the circle has no common point
In this case line PQ is called a non-intersecting line with respect to the circle.
(b) There are two common points A and B that the line PQ and the circle have
In this case line PQ is called a Secant of the circle.
(c) 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
Tangent to a Circle
A tangent to a circle is a line that intersects the circle at only one point.
Or, A line that intersects the circle at only one point is called a TANGENT.
The word 'tangent' comes from the Latin word 'tangere', which means to touch. This word was introduced by the Danish Mathematician Thomas Fineke in 1583.
Properties of a Tangent to the circle
(a) There is only one tangent at a point of the circle.
(b) The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide.
(c) 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.
(d) There is no tangent to a circle passing through a point lying inside the circle.
(e) There is one and only tangent to a circle passing through a point lying on the circle.
(f) There are exactly two tangents to a circle through a point lying outside the circle.
Theorem (1) related to circle
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof:
Let there is a circle with centre O
And there is a tangent XY to the circle at point P.
It is to be proved that OP is perpendicular to XY
Let a point Q on XY other than P
O and Q is joined
The point Q must lie outside the circle because XY is a tangent.
And if Q lies inside the circle, XY will become a secant and not a tangent to the circle.
Thus, OQ is longer than the radius OP.
Thus, OQ > OP
This happens for every point on the line XY except the point P.
Thus, OP is the shortest of all the distances of the point O to the points XY.
Thus, OP is perpendicular to XY.
Conclusion:
(a) At any point on a circle there can be one and only one tangent.
(b) The line containing the radius through the point of contact is also sometimes called the 'normal' to the circle at the point.
Theorem (2) related to circle
The lengths of tangents drawn from an external point to a circle are equal.
Proof:
Let a point P is lying outside the circle having center O
Two tangents PQ and PR are drawn from the point to the circle
Now, it is to be proved that PQ = PR
Construction:
A line OP is drawn from P to the centre O of circle
OQ and OR two lines are drawn from the centre O of the circle
Now, since PQ is a tangent to the circle at point Q and OQ is the radius of the circle
Therefore, PQ is perpendicular to QO
PR is a tangent to the circle at point R and OR is the radius of the circle
Thus, in similar way,
PQ is perpendicular to OR
Thus, ∠ PQO = ∠ PRO = 900
Now, In right triangle OQP and ORP,
OQ = OR
[Because these are radii of the same circle and radii of same circle are equal]
OP = OP
[Common in both the triangles]
∴ Δ OQP ≅ Δ ORP
Thus, PR = PR Proved
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