Circles (వృత్తములు)

A is true: area=πr². R is false: if r doubles, area=π(2r)²=4πr²; area becomes 4 times not 2 times.
A సత్యం. R అసత్యం: r రెట్టింపు→వైశాల్యం 4 రెట్లు.

CD = 2√(r²-36); First find r

r² = 9²+4² = 81+16=97.
CD/2 = √(97-36)=√61. CD=2√61 ≈ 15.62 cm.

Hmm: Let me compute r from AB:
AM=9, OM=4. r²=81+16=97.
CM²=97-36=61. CD=2√61.

PA × PB = PC × PD

Intersecting Chords Theorem:
Circle లో రెండు chords P వద్ద intersect అయినప్పుడు:
PA × PB = PC × PD.

Proof: △PAC ~ △PDB (AA similarity) → PA/PD = PC/PB → PA×PB = PC×PD.
ఉదా: PA=3, PB=8, PC=4 → PD=6. 3×8=24=4×6 ✓

A is true: equidistant chords are equal. R is false: equal chords are equidistant from centre.
A సత్యం. R అసత్యం: సమాన జీవలు కేంద్రానికి సమాన దూరంలో ఉంటాయి.

Central angle = 2 × Inscribed angle

Same arc AB పై:
Central angle ∠AOB = 2 × Inscribed angle ∠APB.

ఉదా: Central angle = 80° → Inscribed angle = 40°.
Special case: Semicircle → Central = 180° → Inscribed = 90° (Thales!)
This is most tested circle theorem!

Chord మరియు arc మధ్య ఉన్న region

Segment = chord తో cut అయిన circle part.
Minor segment = smaller part (minor arc + chord).
Major segment = larger part.

Sector vs Segment:
Sector = Pie slice (2 radii + arc).
Segment = Chord + arc (radii లేదు).
గుర్తు: Segment లో center ఉండదు (minor segment లో)!

Arc = circle boundary యొక్క ఒక భాగం; Minor < semicircle, Major > semicircle

Arc = circle పై రెండు points మధ్య boundary.
Minor arc = smaller part (< 180°). Major arc = larger part (> 180°).
Semicircle = exactly 180°.

గుర్తు: AB chord రెండు arcs చేస్తుంది — minor (చిన్నది) మరియు major (పెద్దది)!

∠C=105°, ∠D=100°

Opposite angles supplementary:
∠A+∠C=180° → ∠C=105°.
∠B+∠D=180° → ∠D=100°.
Check: 75+80+105+100=360° ✓

6√2 cm

Square inscribed in circle: diagonal = diameter = 12.
Side = diagonal/√2 = 12/√2 = 6√2 cm.

Area of square = side² = (6√2)² = 72 cm².

Circle పై రెండు points ని కలిపే line segment

Chord = circle యొక్క boundary పై రెండు points ని కలిపే line segment.
Diameter = longest chord (center గుండా వెళ్ళే chord).

ఉదా: AB ఒక chord అయినప్పుడు A మరియు B circle పై ఉంటాయి.
గుర్తు: Chord ⊂ inside of circle; Secant = chord ని extend చేసిన line!

8 cm

r=10. Half chord=6.
d=√(10²-6²)=√(100-36)=√64=8 cm.

Central angle for semicircle = 180°. Inscribed = half = 90°.

AB = diameter → Central angle ∠AOB = 180°.
Inscribed angle theorem: Inscribed angle = ½ × Central angle.
∠APB = 180°/2 = 90°. ✓

ఇది Thales theorem యొక్క proof!

8 cm

AM² = r²-d² = 25-9=16 → AM=4.
Chord = 2×4 = 8 cm.

3-4-5 Pythagorean triple!

77/3 ≈ 25.67 cm²

Sector area = (θ/360°) × πr².
= (60/360) × (22/7) × 49.
= (1/6) × 154 = 77/3 ≈ 25.67 cm².

Formula గుర్తు: Sector area = (θ/360) × πr².
Full circle θ=360° → Area=πr² (check ✓)!

r = 11 cm

Arc length = (θ/360°)×2πr.
22π/3 = (120/360)×2πr = (1/3)×2πr.
22/3 = 2r/3 → 2r = 22 → r=11 cm.

Supplementary (మొత్తం 180°)

Cyclic quadrilateral (circle లో inscribed quadrilateral):
Opposite angles మొత్తం = 180°.
∠A + ∠C = 180°, ∠B + ∠D = 180°.

ఉదా: Rectangle circle లో ఉంటుంది → 90°+90°=180° ✓.
ఇది circle + quadrilateral topics రెండింటిలో వస్తుంది!

4 common tangents

Two circles external గా (d > r₁+r₂) అయినప్పుడు 4 common tangents.
d=10, r₁+r₂=8. 10>8 → External circles.
4 tangents: 2 direct + 2 transverse.

Cases:
d > r₁+r₂ → 4 tangents (external)
d = r₁+r₂ → 3 tangents (externally tangent)
r₁-r₂ < d < r₁+r₂ → 2 tangents (intersecting) d = r₁-r₂ → 1 tangent (internally tangent) d < r₁-r₂ → 0 tangents (one inside other)

Triangle circumcircle పై point P నుండి మూడు sides కి feet collinear అయినప్పుడు ఆ line Simson line

Simson Line Theorem:
Triangle ABC circumcircle పై point P నుండి BC, CA, AB కి perpendiculars వేస్తే feet D, E, F collinear → DEF = Simson line.

Converse: D,E,F collinear → P circumcircle పై ఉంటుంది.
Elegant theorem connecting circles and lines!

r = 5

PA=PB=5√3, ∠APB=60° → ∠APO=30° (OA⊥PA).
tan(∠APO) = OA/PA... wait: ∠OPA=30°.
sin(∠OPA)=OA/OP... but OA⊥PA.
tan(30°)=OA/PA=r/(5√3).
1/√3 = r/(5√3) → r=5.

Check: OP=√(r²+PA²)=√(25+75)=10. ∠OPA=arcsin(5/10)=30° ✓

Power = OP² - r² = PT² = PA×PB

Power of a Point:
P outside circle:
Power = OP²-r² = PT² (tangent) = PA×PB (secant).

P inside circle:
Power = -(r²-OP²) = negative.
PA×PB = PC×PD (intersecting chords — power magnitude).

గుర్తు: Power of a point unifies all circle theorems!

50°

∠AOB = 2×∠ACB = 80° (central=2×inscribed).
OA=OB (radii) → △OAB isosceles.
∠OAB = ∠OBA = (180°-80°)/2 = 50°.

r = 7 cm

(90/360)×(22/7)×r² = 38.5.
(1/4)×(22/7)×r² = 38.5.
r² = 38.5×4×7/22 = 49 → r=7.

OM = 5 cm

AM = 12 (half chord).
OM² = OA²-AM² = 169-144=25 → OM=5 cm.

5-12-13 triple!

External center of similitude — r₁:r₂ గా external గా divide చేస్తుంది

External common tangents → External center of similitude (S).
S divides line O₁O₂ externally in ratio r₁:r₂.

Internal tangents → Internal center of similitude.
Internally divides O₁O₂ in r₁:r₂.

These are homothety centers!

Triangle exterior కి tangent, రెండు sides extensions కి tangent అయిన circle

Excircle opposite to A (I_A):
✦ Side BC కి externally tangent.
✦ AB మరియు AC extensions కి tangent.
✦ Internal angle bisector A మరియు external bisectors B,C కలిసే point = I_A.

Excircle radius r_A = Δ/(s-a).
(Δ=area, s=semi-perimeter, a=BC)
ప్రతి triangle కి 3 excircles!

ఒకే center ఉన్న వేర్వేరు radius గల circles

Concentric circles = same center కానీ different radii.
ఉదా: Target board (archery) — అన్ని circles same center.
Ripples in water = concentric circles!

గుర్తు: "Con" = together, "centric" = center — same center circles.

70°

Central angle = 2 × Inscribed angle.
= 2 × 35° = 70°.

Center=(2,3), r=2

Complete square:
(x²-4x+4)+(y²-6y+9) = -9+4+9 = 4.
(x-2)²+(y-3)²=4.
Center=(2,3), r=2.

r₁² = r₂² + r₃² ... wait, let me think

Outer circle radius R, middle r, inner touches chord at center.
Chord AB of outer (radius R) tangent to middle (radius r):
Distance from center to chord = r.
If chord passes through inner center (radius r₃=0, just a point), and inner circle has center on chord...

Actually: outer radius = R, middle = r, and inner center = origin.
Chord tangent to middle → distance to center = r.
Chord passes through inner center (at origin) → distance = 0.
Contradiction unless inner = degenerate.

Standard: Chord of outer tangent to middle. Half chord = √(R²-r²).
This is the standard concentric circle problem.
R² = r² + (half chord)².

C = 2πr (లేదా πd)

Circumference = circle యొక్క perimeter (boundary length).
C = 2πr = πd.
π ≈ 3.14159... (irrational number).

ఉదా: r=7 → C = 2×(22/7)×7 = 44 cm.
గుర్తు: π = 22/7 approximation exam లో చాలా use అవుతుంది!

Normal: y/x = y₁/x₁ లేదా xy₁ = yx₁

Normal at P(x₁,y₁) to circle x²+y²=r²:
Normal passes through center (0,0) and P(x₁,y₁).
Equation: y/x = y₁/x₁ → xy₁-yx₁=0.

Normal always passes through center of circle!
This is because radius ⊥ tangent at P, and normal = radius direction.

2:1

Central angle : Inscribed angle = 2:1.
Same arc పై central = 2× inscribed.

ఉదా: Arc AB subtends 80° at center → 40° at any point on remaining circle.

Common chord పై ప్రతి point నుండి రెండు circles కి equal power

Common chord AB పై point P:
Power w.r.t. circle 1: PA×PB (chord P వద్ద).
Power w.r.t. circle 2: PA×PB (same, since A,B on both circles).
Equal power → P on radical axis.
కాబట్టి common chord ⊆ radical axis.
Both perpendicular to line of centers → they coincide!

220°

Inscribed ∠B=70° → minor arc AC = 2×70° = 140°.
Major arc AC = 360°-140° = 220°.

r = 10 cm

Perpendicular from center bisects chord.
AM = 8 cm (half of chord).
OA² = OM² + AM² = 36+64 = 100.
r = OA = 10 cm.

Diameter = 2 × Radius

Circle లో center నుండి boundary కి ఉన్న దూరం = Radius (r).
ఒక end నుండి center గుండా మరొక end కి = Diameter (d).
d = 2r లేదా r = d/2.

గుర్తు: Diameter అంటే circle గుండా వెళ్ళే longest chord!
ఉదా: r=5 → d=10; d=14 → r=7.

Nine-point circle, incircle ని internally tangent గా touch చేస్తుంది; excircles ని externally

Feuerbach's Theorem (1822):
Nine-point circle:
✦ Incircle ని internally tangent గా touch చేస్తుంది.
✦ మూడు excircles ని externally tangent గా touch చేస్తుంది.

Feuerbach point = nine-point circle మరియు incircle tangency point.
One of the most beautiful theorems in triangle geometry!

True — Equal chords equidistant from center

ఒకే circle లో:
Equal chords → Center నుండి equal distance.
Converse కూడా true: Equal distance → Equal chords.

Proof: Perpendicular from center to chord bisects it.
Equal chords → equal half-lengths → equal distances (Pythagoras).

OT = 5 cm

OT ⊥ PT → OT²+PT²=OP².
OT²+144=169.
OT²=25 → OT=5 cm.

5-12-13 Pythagorean triple! గుర్తు: tangent problems లో Pythagoras use చేయాలి.

56 cm²

Sector area = (90/360)×πr² = ¼×(22/7)×196 = ¼×616=154 cm².
Triangle area = ½×r×r×sin90° = ½×14×14=98 cm².
Segment = Sector - Triangle = 154-98=56 cm².

Area = 616 cm²; Circumference = 88 cm

Area = πr² = (22/7)×14² = (22/7)×196 = 22×28 = 616 cm².
Circumference = 2πr = 2×(22/7)×14 = 2×44 = 88 cm.

గుర్తు: r=14 (14 is 2×7) — 22/7 తో easily calculate అవుతుంది!

PT² = PA × PB

Tangent-Secant Theorem:
PT² = PA × PB.

Proof: △PAT ~ △PTB (AA) → PT/PB = PA/PT → PT² = PA×PB.
గుర్తు: Tangent is geometric mean of whole secant and external segment!
ఉదా: PA=4, PB=9 → PT=√36=6.

∠AOB = 140° (minor) లేదా 220° (major reflex)

∠P = ½|arc₁ - arc₂| (tangent-tangent angle formula).
OAPB quadrilateral: 90°+90°+40°+∠AOB=360°.
∠AOB=140°.

Or: ∠P+∠AOB=180° (for two tangents from external point).
40°+140°=180° ✓

Same arc పై (same segment లో) inscribed angles సమానంగా ఉంటాయి

Theorem: Same chord AB యొక్క same side లో ఉన్న circle పై అన్ని inscribed angles equal.
∠APB = ∠AQB (P, Q same segment లో ఉంటే).

ఉదా: ఒక stadium చుట్టూ ఉన్న అన్ని spectators కి same chord "football goal" same angle లో కనిపిస్తుంది!

P'=(0,8)

OP = 2. OP×OP'=16 → OP'=8.
P' same direction: P'=(0,8).

Circle inversion maps inside↔outside.
Lines not through O → circles; circles through O → lines!

Both true. C=2πr. R shows d=2r so πd=2πr.
A, R రెండూ సత్యం. C=2πr.

6 cm

AM = 8 (half chord). OA=10 (radius).
OM² = OA²-AM² = 100-64 = 36 → OM=6 cm.

Tangent-chord angle = inscribed angle in alternate segment

Theorem (Alternate Segment Theorem / Tangent-Chord Angle):
Tangent PQ మరియు chord AB వద్ద angle = alternate segment లో inscribed angle.

∠PAB = ∠ACB (C alternate segment లో).
ఇది "Tangent-chord angle = angle in alternate segment"!
Exam లో చాలా important theorem.

50°

PA=PB (equal tangents) → △PAB isosceles.
∠PAB=∠PBA=(180°-80°)/2=100°/2=50°.

l² + d² = r²

Perpendicular bisects chord.
Half chord = l, radius = OA = r, distance = d = OM.
Right triangle OMA: l²+d²=r².

ఇది standard relation: l=√(r²-d²), d=√(r²-l²).