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Sunday, 22 December 2024

1-IMP 2025 CLASS 10 MATHS CIRCLE CHAPTER 10

 

Prove that the tangents drawn at the ends of a diameter of a circle are parallel
Proof:
Important Questions for Class 10 Maths Chapter 10 Circles 27
∠1 = 90° …(i)
∠2 = 90° …(ii)
∠1 = ∠2----by (1) &(2) (AIA) 
∴PQ || RS

Q 2. Prove that the length of the tangents drawn from an external point to a circle are equal.
Answer:
Given :

Let AP and BP be the two tangents drawn from external point P to the circle with centre O.

To Prove : AP = BP


Proof :


In Δ AOP and Δ BOP


OA = OB (radii) 


OAP=OBP=90 


(Line drawn from from center to the tangent through the point of contact is perpendicular) 


OP = OP (common)


ΔAOP ΔBOP (R.H.S.) 


 AP = BP ( C P C T ) 


Therefore the length of the


 tangents drawn from an


 external point to a circle are


 equal.


Q 3. Statement: 

BPT (Basic Proportionality Theorem), 

If a line is drawn parallel to one of the  triangle to intersects the other two sides in distinct points, 

then the other two sides of 

the  triangle are divided  into the same ratio. 


Proof:

Given:


In ∆ABC, DE || BC and AB and AC are intersected by DE at points D and E respectively.


To prove:


AD / DB = AE / EC


Construction:

Join BE and CD.and

Draw:

EGAB and DFAC

Proof:

We know that

ar( Δ ADE) = 1 / 2 × AD × EG   

ar( Δ DBE) = 1 / 2 × DB × EG   

So

ar(Δ ADE) / ar(Δ DBE) =

 AD / DB   ------- (1)


Similarly,


ar(Δ ADE) / ar(Δ ECD) =

 AE / EC   ----------(2)

Now, 

Δ DBE and Δ ECD

are the on the same base DE and also between the same parallels i.e. DE and BC,

So

ar(Δ DBE) = ar(Δ ECD)  ---(3)


By (1), (2) , (3) 


AD / DB = AE / EC  


Hence  proved.




Q 4.

Prove √3 is a Irrational number.

Proof:

Let √3 be a rational number.


So,  3=p ____ (1) q       

On squaring both sides

3=p2q2
q2= p23

3 is a factor of p2

3 is a factor of p.

Now, again let p = 3 c.

So,  3= 3 cq

On squaring both sides



3
= 9 c2q2




q
2
=3 c2






c
2
= q23




3 is factor of 
q2



3
 is a factor of q.

Here 3 is a common factor of p, q  both

 So p, q are not co-prime.

Therefore our assumption is wrong. 

3  is an irrational number.

Q 5.

Prove that the parallelogram circumscribing a circle is a rhombus



 ABCD is a parallelogram  , So opposite sides of a parallelogram are equal. 

AB=CD.

BC=AD.-----++++(1)

DR=DS --------------(2) 

 (Tangents on the circle from same point D)
Similarly

CR=CQ ----------(3)

AP = AS ------------(4) 

PB = BQ  -------++(5) 

Adding all these equations we get

DR+CR+BP+AP=DS+CQ+BQ+AS

(DR+CR)+(BP+AP)=(DS+AS) +( BQ + CQ) 

CD+AB=AD+BC

But   AB =CD   &   BC=AD

     AB + AB = BC + BC

2AB=2BC

Therefore AB=BC -------(6)

From equation (1)& (6) we get

AB=BC=CD=DA

ABCD is a Rhombus

Q 6. Find relationship between x and y such that the point P(x, y) is equidistant from the points A (2, 5) and B (-3, 7)

Solution:

Let P (x, y) be equidistant from the points A (2, 5) and B (-3, 7). so

∴ AP = BP

         AP 2 = BP 

(x – 2)2 + (y – 5)2 = (x + 3)2 + (y – 7)2

x2 – 4x + 4 + y2 – 10y + 25
                 =
x2 + 6x + 9 + y2 – 14y + 49

⇒ -4x – 10y +29 =  6x - 14y +58 

⇒ -10x + 4y = 29

 is the required relation.

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