Crazytoppers Free Education: Circle

Circles

Circles are closed plane curves with all points on the curve equally distant from a fixed point called the

center. The symbol Q indicates a circle. A circle is usually named by its center.

A line segment from the center to any point on the circle is called the radius (plural, radii). All radii of the same circle are equal

Circumference

The circumference of a circle is the distance around the circle. The value of the circumference is equal to 2rrr where r is the radius of the circle.

Diameter

The diameter of a circle is a chord that passes through the center of the circle. The diameter, the longest distance between two points on the circle, is twice the length of the radius. A diameter perpendicular to a chord bisects that chord. In the figure below

C is the center.

DCE and FCG are diameters.

AB ┴ DCE, so AP = PB

Lines of circle

Lines like m are called tangents. A tangent is a line that has one of its points on a circle and the rest outside the circle.

Line n is called a secant of the circle. A secant is defined as any line that intersects a circle in two distinct points.

A secant whose end points lie on a circle is called a Chord. In figure above AB is a chord of the circle. Thus, a chord is always a part of secant.

Chord

Chord is a line segment whose endpoints are on the circle. In the figure AB, FG and DC are the chords. A circle can have an infinite number of chords of different lengths.

El. A, B, C & D lie on a circle with center P. Classify the following segments as radii and chords.

PA, AB, AC, BP, DP, DA, PC, BC, BD, CD.

Sol. Chords : AB, AC, DA, BC, BD, CD.

Radii : PA, BP, DP, PC.

Central Angle

A central angle is an angle whose vertex is the center of a circle and whose sides are radii of the circle. An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle.

L ACB is a central angle

Inscribed angles

Whereas central angles are formed by radii, inscribed angles are formed by chords.

As shown in above figure, the vertex S of the inscribed angle RST is on the circle.

Arc

An arc is a portion of a circle. The symbol half circle is used to indicate an arc. Arcs are usually measured in degrees. Since the entire circle is 360⁰, a semicircle (half a circle) is an arc of 180° and a quarter of a circle is an arc of 90°.

ABD, AB and BD are the arcs.

AB is called the minor arc and ADB is the major arc. The minor arc is always represented by using the two end points of the arc on the circle. However, it is customary to denote the major arc using three points. The two end points of the major arc and a third point also on the arc. If a circle is cut into two arcs such that there is no minor or major arc but both the arcs are equal then each arc is called a semicircle.

Measure of an arc

The measure of a semicircle = 180°.

The measure of a minor arc is equal to the measure of its central angle.

m(arc RQT) = m∟RCT

The measure of a major arc = 360° — (measure of corresponding minor arc)

m(arc RST) = 360° — m(arc RQT)

Intercepted Arc

An arc is said to be intercepted by an angle if each side of the angle contains an endpoint of the arc and the arc but for its endpoints, lies in the interior of the angle.

Arc BD and arc AC are intercepted by the LAOC.

E2. a) In the figure given below name the central angle of arc AB.

b) In the figure what is the measure of arc AB.

c) Name the major arc in the figure.

Sol. a) LAOB

b) 80°. The measure of an arc is the measure of its central angle.

c) Arc AXB

E3. a) In the figure below name the inscribed angle and the intercepted arc.

b) What is m (arc PQ)?

Sol. a) inscribed angle — ∟ PRQ ,intercepted arc — arc PQ

b) 60°. The measure of an intercepted arc is twice the measure of its inscribed angle.

E4. LPAQ and LPBQ intercept the same arc PQ. What is the m LPBQ and m (arc PQ)?

Sol. m ∟PBQ = 40°. If two inscribed angles intercept the same arc, their measures are equal

m (arc PQ) = 80° as m (arc) = 2m (inscribed angle).

Important

Ø Only one circle can pass through three given points.

Ø There is one and only one tangent to the circle passing through any point on the circle.

Ø From any exterior point of the circle, two tangents can be drawn on to the circle.

Ø The lengths of two tangents segment from the exterior point to the circle, are equal

Ø The tangent at any point of a circle and the radius through the point are perpendicular to each other.

Ø When two circles touch each other, their centres & the point of contact are collinear.

Ø If two circles touch externally, distance between centres = sum of radii.

Ø If two circles touch internally, distance between centres = difference of radii

Ø Circles with same centre and different radii are concentric circles.

Ø Points lying on the same circle are called concyclic points.

Ø Measure of an arc means measure of central angle. m(minor arc) + m(major arc) = 360⁰.

Ø Angle in a semicircle is a right angle.

Alternate Segment Theorem

In the figure given below if BAC is the tangent at A to a circle and if AD is any chord, then

LDAC = LAPD or LPAB = LPDA

(Angles in alternate segment)

Theorems on Circles

If If ON is ┴ from the centre 0 of a circle to a chord AB, then AN = NB.

(┴ from centre bisects chord)

Ø N is the midpoint of a chord AB of a circle with centre 0, then LONA = 90°.

(Converse, ┴ from centre bisects chord)

(Eq. chords are equidistant from centre)

E5. In a circle with radius 5cm, a chord is drawn at the distance 3cm from the centre. Find the length of the chord.

SOL. let O be the centre of the circle and AB is the chord . in rt. Δ AMO , by pythagorus theorem

AO² = AM² + OM²

= 5² = AM² + 3²

AM2 = 25 – 9 = 16

AM = 4

We know that perpendicular from the centre bisect the chord.

So, AB = 2 x AM = 2 x 4 = 8cm.

Ø Equal chords in a circle are equidistant from the centre. If the chords AB and CD of a circle are equal and if OX┴AB, OY┴CD then OX = OY.

Ø If OX ┴ chords AB , OY ┴ chord CD and OX = OY, then chords AB = CD (Refer the figure given above).

(Chords equidistant from centre are eq.)

Ø A circle with centre 0, with ∟AOB at the centre, LACB at the circumference, standing on the same arc AB, then LAOB = 2∟ACB

(L at centre = 2L at Circumference on the same arc AB)

E6. Find the value of x in the following circle with centre 0

Sol. We know that angle at centre is equal to the twice of the angle at circumference by the same chord or arc. So, 108= 2x or x = 54⁰

Ø In a circle with centre 0, LACB and ∟ADB are the angles at the circumference, by the same arc AB, then LACB = LADB.

(Ls in same arc or Ls in same seg.)

E7. Find the value of y in the adjoining circle with centre 0.

Sol. Clearly 120⁰ = 2x or x = 60°

Since the angles in same arc or segment are equal, so y = x = 60°.

E8. As shown in the figure, triangle PAB is formed by three tangents to circle with centre 0 and LAPB = 40°. ∟AOB equals

Sol. AT=AS, OS=OT

BS=BR,OS=OR

Therefore OA and OB bisect angles SOT and SOR

Now, LORP = LOTP = 90° and LAPB = 40° ∟ROT = 140°

2 ( a+b) = 140°

∟AOB = (a+b) = 70⁰. Ans.

Ø In a circle with centre O and diameter AB , if C is any point at the circumference , then ∟ACB = 90⁰.

(L in semi-circle)

Ø If LAPB = LAQB and if P, Q are on the same side of AB, then A, B, Q, P are concyclic.

(AB subtends equal Ls at P and Q on the same side)

Ø In equal circles (or in the same circle) if two arcs subtend equal angles at the centre (or at the circumference), then the arcs are equal.

Ø If LBOA = LXOY, then arcAB = arc XY or if LBPA = LXQY, then

arcAB = arc XY.

(Eq. Ls at centre or at circumference stand on equal arcs)

Ø In the same or equal circles if chords AB = CD, then arc AB = arc CD.

(Eq. chords cut off equal arcs)

Few important results

Ø If two chords AB & CD intersect externally at P then

ü PA x PB=PC x PD

ü LP = ½ [m (arc AC) — m (arc BD)]

Ø If PAB is a secant and PT is a tangent then ,

ü PA x PB = PT²

ü LP = 1/2[m(arc BXT) — m(arc AYT)]

E9. In the figure, if 0 is the centre of the circle, then LBTC = ?

Sol. Join BC

∟ABC = LACS = 67°

(Angle made by tangent ST to a circle with a chord drawn from the point of contact is equal to the angles

in the alternate segment of a circle)

Also, LACB = 90⁰

(Angle inscribed in semicircle is a right angle)

180° = 67⁰ + 90° + LBCT; LBCT = 23°

In ΔBCT, ∟BCT = 23°, LCBT = 113°

(: LABC + LCBT = 180°); LBTC = 180° — LCBT — LBCT = 180° — 113° — 23° = 44⁰.

Ø If chords AB & CD intersect internally at P, then

ü PAxPB=PCxPD

ü LBPD = 1/2[m(arc AXC) + m(arc BYD)]

E10.In the given figure, m ∟EDC = 54°, m ∟DCA = 40°. Find x,

y and z respectively.

(1) 20°, 27°, 86° (2) 40°, 54°, 86°

(3) 20°, 27°, 43° (4) 40°, 54°, 43°

Sol. m LACD = m ∟DEC

mLDEC=x=40°

m LECB = m LEDC

m LECB = y = 54°

54⁰ + x + z = 180° .... (sum of all the angles of a triangle)

54° + 40° + z = 180°

:. z = 86°. Ans.(2)

E11.Two tangents of length 21 inches from a point P to the circle with centre 0 are inclined at an angle of 60°. Find

the circumference of the circle. (Taken = Pie = 22/7 )

Sol. ΔAOP = ΔBOP ... (SSS test of congruence)

m ∟APO = m LOPB = 30°

ΔOAP is a 30° — 60° — 90° triangle.

Side opposite to 60° = √3/2 of hypotenuse

21 = 3^1/2/2 x OP

..OP = 42/3^1/2

OA=1/2 xOP = ½ x 42/3^1/2= 21/3^1/2

Circumference = 2 x 22/7 x 21/3^1/2 = 44√3 inches.

E12.In the figure given below, AQ is 6 cm in length, PQ is 18 cm in length and is a chord of the circle with centre 0 and AC is the tangent. Find the length of the tangent.