1. Name- NITI VERMAN, PRAHALAD, SHUBHAM, VIKAS KR. PASWAN, SUMIT
MATH PROJECT WORK
Name- NITI VERMAN, PRAHALAD, SHUBHAM, VIKAS KR. PASWAN, SUMIT
CLASS-X A

2. CIRCLES
A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. 
SECANT- A line which intersects a circle in two distinct points in called a secant.
TANGENT – A Line meeting a circle only in one point is called a tangent to the circle at that point.
Secant
Tangent

3. There is no tangent passing through a point lying inside the circle
NUMBER OF TANGENTS TO A CIRCLE
There is no tangent passing through a point lying inside the circle
There is one and only one tangent passing through a point lying on a circle.
There are exactly two tangents through a point lying outside a
circle.
LENGTH OF TANGENT
The length of the line segment of the tangent between a given point and the given point of contact with the circle is called the length of the tangent from the point to the circle.

4. RESULTS ON TANGENTS
THEOREM 1 The tangent at any point of a circle is perpendicular to the radius through the point of contact.
GIVEN A Circle with centre O and a tangent AB at a point P of the circle. . O
TO PROVE OP AB.
CONTRCUTION Take a point Q, other than P, on AB. Join OQ.
PROOF:- Q is a Point on the tangent AB, other than the point of contact P.
Hence- Q Lies outside the circle.
Let OQ intersect the circle at R.
Then, OR < OQ [ a part is less than the whole.] (i)
But, OP= OR [ radii of the same circle.] (ii)
Hence- OP< OQ [ from (i) and (ii) R Q A P B

5. Thus, OP shorter than any other line segment joining O to any points of AB, other than P. But , the shortest distance between a point and a line is the perpendicular distance. OP AB Proved

6. THEOREM 2 (Converse of Theorem 1) A Line drawn through the end of a radius and perpendicular to it is tangent to the circle.
GIVEN A Circle with centre O in which OP is radius and AB is a line through P such that OP AB. .
TO PROVE AB is tangent to the circle at the point P. P Q A B
CONSTRUCTION Take a point Q, different from P, on AB. Join OQ.
PROOF We Know that the perpendicular distance from a point to a line is the shortest distance between them. .
. .
OP AB= OP is the shortest distance from O to AB.
OP< OQ
Q Lies outside the circle
AB meets the circle at the point at P only
Hence AB is the tangent to the circle at the point P. .
. .

7. END