Theorems No Circle Class 10 Ch-10

PROOF OF IMPORTANT THEOREMS ON THE CIRCLE

Coplete explanation with proof of Theorem 10.1 and  theorem 10.2 class 10 chapter 10, strictly based on NCERT Book and syllabus.

THEOREM 10.1 CHAPTER 10 CLASS 10

Statement :

Tangent is always perpendicular to the radius at the point of contact.

Given: In circle C(o,r), XY is tangent to the circle at point P.

To Prove:  OP ⊥ XY

Construction : 

Take any arbitrary point Q (other than P ) on the line XY and join OQ which meet the circle at point R.

Proof:

In order to prove that OP ⊥ XY it is sufficient to prove that OP is the smallest line segment than all the line segments  obtained by joining O with any point on XY.

OP = OR .......... (Equal radii)

Now OQ = OR + RQ

OQ = OP + RQ

Subtract RQ from the R.H.D. we get

OQ > OP or

OP < OQ

But Q is an arbitrary point on XY

⇒ OP is the smallest line segment and smallest line segment is always perpendicular.

Hence OP ⊥ XY

Hence prove the required theorem

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THEOREM 10.2  CHAPTER 10 CLASS 10

Statement

Prove that length of tangents from external point to the circle are equal in length.

Given : AC and BC are two tangents from external point to the circle.

To Prove : AC = BC

Construction:  Join OA, OB and OC

Proof:

Since radius is always perpendicular to the tangent.

∴ ㄥ1 = ㄥ2 = 90^o

In ΔAOC and ΔBOC
OA = OB ............... (Equal radii)
OC = OC .............. (common side)
ㄥ1 = ㄥ2 .............. (Each = 90o 
∴ By RHS ≌ rule
ΔAOC ≌ ΔBOC 
⇒ AC = BC .............By CPCT
Hence prove the required result 

Results of other important theorems on circle

Theorem : Angle made by the chord of the circle with the tangent of the same circle (at the point of contact of the chord and tangent) is equal to the angle made by the chord in the alternate segment of the circle. 

Theorem : If AB and CD are two chords of the circle intersect each other at point p inside or outside  the circle then:- 

PA X PB = PC X PD

This result rmain same in both the cases inside and out side the circle.

When Chord intersect each other  Inside the circle

        

When chords intersect each other Outside the circle

    

Theorem : If chord AB of a circle intersect the tangent of the same circle at point P outside the circle then 

PA X PB = PT²

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