Line and circle

A line just touches the circle is called tangent of the circle.In the picture Red Line is a tangent to the circle.

A) Prove that the tangent at a point on a circle isperpendicular to the diameter through that point.

Proof

Let O = x0

OAB is isosceles and A =

when B approaches A ( Click here for Video-Number 0 )x becomes 0Then A = 90-0 = 90.

B . Draw this picture in your notebook.

Drawing first picture – CLICK HERE -VIDEO -1

Drawing Second picture -CLICK HERE -VIDEO -2

Answer:

click here-video-3

Answer: Rhombus

Answer:Tangents are drawn at the ends of of a diameter.Alternate angles are 90 degree.That means these tangents are parallel.

Tangents and angles

1. Prove that OBCA is cyclic.Answer:AC and BC are tangents.

OAC = OBC =900

and sum 1800.

ie O+ C = 1800.

Opposite angles angles are supplementary.Ie OBCA is cyclic.

2. The circle in the picture is a unit circle.Find Tan x .

Answer:

Tan x = = = = PT

ie

Tan of central angle of a unit circle is the Tangent itself.

C C

C C

3. Draw a circle of radius 3 cm and an equilateral triangle with sides touching the circle.

Answer:

See video 4

Now Try following problems

Answer:Angle of this triangle are 40,60,80That is central angles are 140, 120, 100

see this video- 5

Answer:

i )Since small triangle is equilateral all angles 60 degreeThat is central angle is 120 degreeie each angle of big triangle is 60 and there fore anequilateral triangle.

ii)

Step 1: draw an equilateral triangle with sides 3 cmsStep 2: Find its cir-cum centre by drawing perpendicular bisectors to each side.

Draw the cir-cum circle.

Step 3 : Draw radii to vertices of triangle and draw perpendiculars to all radii to construct bigger triangle. Its side will be 6 cm

Iii)

Let angles of inner triangle are x,y,z where x≠y≠zThen central angles are 2x ,2y,, 2z ( Central angle is twice of angle inscribed in same arc )Then angles of bigger triangle = 180-2x, 180-2y, 180-2z180-2x ≠ 180-2y ≠ 180-2zand angles of both triangles different.That is they are not similar.

Answer:

i ) Consider triangles APO and BPO

= =900.OP is a common sideOA and OB are radii of same circle and equal.

≡

ie AP = BP , Tangents are equal.

Ii )

Consider triangles APO and BPOAll sides are equalthere fore

≡

III )

Consider Triangle APQ and BPQproved

and AP = BPChord and tangentPQ is a common sideTwo sides and included angles are equal.

≡

AQ= BQ

and they are linear pairs.

So PO is perpendicular bisector of AB

Answer:

Given that PR and QS are perpendicular chords.OS, OP, OR and OQ radii perpendicular to the tangents

Let Then because angle between the tangents and angle between radii to point of contact are supplementary.

Let

Then

Now

To prove

In the II figure let PR perpendicular to SQCentral angle of arc SR = x0.Central angle of arc PQ =y0.Given

Given

so in above figure SOR + POQ = 1800.

180-d +180-b =180

ie ABCD is a cyclic quadrilateral.Chord and tangent

Prove that In a circle, the angle between a chord and tangent at either endis half the central angle of the chord.

Answer:

Let AOB = x0. Is isosceles. (radii are equal )

Base angles are

(angle between diameter and tangent )

which is half of central angle of chord AB.

Answer:

Problems 2, 3, 4 try yourself.

A tangent from outside1. How many tangents can be drawn from a point outside the circle to the circle ?From a point outside the circle we can draw two tangents to the circle

In the picture PA and PB are tangents.

2. Draw tangents to the circle with centre O and radius 2cm from a point outside the circle of distance from the centre 5cm

See video Number 6

3. Prove that ,the tangents to a circle from a point are of the same length.

Answer:Let PA and PB are tangents fromthe circle centred at O

Consider triangles PAO and PBO

Both are right triangles .

So PA = and PB =

ie PA = PB , Tangents are equal.

4. Prove That in a quadrilateral formed by the tangents at four points on acircle, the sum of the opposite sides are equal.

Answer:

Sum of opposite sides =

(a+b) +(c+d) = a+b+c+dand(a+d)+ (b+c) = a+b+c+dsum of opposite sides are equal.

5. In the figure figure PC is a tangent to the circle, Then prove thatPA X PB =PC2.

Answer:

Let PCA = x0.Then AOC = 2x0.(The angle between a chord andtangent at either endis half the central angle of thechord.)

Consider triangles PAC and PBCP is common and PCA = PBC

PAC ~ PBC sides opposite to equal angels are proportional.

C

=

PA X PB = PC2.

Answer:

Perimeter of the triangle =

PC + CD + PD = PC + (CE+ED) + PD

= PC +(BC+ DA) +PD ( Tangents drawnfrom an out side point are equal )

= (PC +BC) + ( DA + PD)= PB + PA

Now OA and OB radii perpendicular to thetangents PA and PB

90 (given )then 90

all angles 90 degree two adjacent sides equal.

PAOB is a square. Ie PB +PA = OB +OA =diameter of circle.