Circles (వృత్తములు)

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!

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.

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.

64π cm²

Ring area = π(R²-r²) = π(100-36) = 64π cm².
= 64×3.14 ≈ 201 cm².

OM = 5 cm

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

5-12-13 triple!

OS = 9 cm

PO×OQ = RO×OS.
3×12 = 4×OS.
36 = 4×OS → OS=9 cm.

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.

8 cm

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

3-4-5 Pythagorean triple!

మూడు circles యొక్క pairwise radical axes కలిసే point

Radical Center:
మూడు circles C₁, C₂, C₃.
Radical axis(C₁,C₂), Radical axis(C₂,C₃), Radical axis(C₁,C₃) → అన్నీ ఒక point వద్ద కలుస్తాయి.
ఆ point = Radical center.

Radical center నుండి మూడు circles కి equal power!

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)

Both true. Angle in semicircle=90° (Thales' theorem). R correctly names it.
A, R రెండూ సత్యం. అర్ధవృత్తంలో కోణం=90°.

10 cm (opposite sides) లేదా 0 cm (same side)

Equal chords → equidistant from center → same distance d from center.
If on same side: distance between chords = 0 (they coincide if equal distance).
If opposite sides: distance = d+d = 2d.

With r and chord=8: half=4. d=√(r²-16).
If centers 5 apart is given differently... Two circles: radius same.
With given info: chords same circle, equal → distance from center = d.
Between chords = 2d (opposite) or 0 (same side).

∠P = ½|arc₁ - arc₂| (difference of intercepted arcs)

External angle formula:
Angle at external point = ½ × (difference of intercepted arcs).

Applies to: 2 secants, 2 tangents, 1 tangent+1 secant.
Compare: Internal angle = ½×(sum of arcs).
External angle = ½×(difference of arcs).
గుర్తు: Inside = sum, Outside = difference!

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.

Both true. Cyclic quad opposite angles=180°. R correctly identifies it as fundamental property.
A, R రెండూ సత్యం. చక్రీయ చతుర్భుజం ప్రాథమిక లక్షణం.

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° ✓

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!

2:1

Area ∝ r². Area ratio = 4:1 = r₁²:r₂².
r₁/r₂ = √4 = 2. Ratio = 2:1.

అవుతుంది (3²+4²=25); Tangent: 3x+4y=25

3²+4²=9+16=25 ✓ → point on circle.
Tangent at (x₁,y₁) to x²+y²=r²: xx₁+yy₁=r².
Tangent at (3,4): 3x+4y=25.

Standard formula: Tangent at (x₁,y₁) to circle x²+y²=r² is xx₁+yy₁=r².

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.

Circle ని సరిగ్గా ఒక point వద్ద touch చేసే line

Tangent = circle ని exactly one point వద్ద touch చేసే line.
ఆ point ని "point of tangency" అంటారు.

Tangent ఎల్లప్పుడూ radius కి perpendicular గా ఉంటుంది!
ఉదా: టైర్ రోడ్ ని touch చేసే point — road tangent to tire circle!

Arc length = (θ/360°) × 2πr

Arc length = (θ/360) × 2πr = (θ/360) × circumference.
θ = angle subtended at center.

ఉదా: θ=90°, r=7 → Arc = (90/360)×2×(22/7)×7 = ¼×44 = 11 cm.
గుర్తు: Full circumference θ=360° → L=2πr ✓.

Center=(-D/2, -E/2); r=½√(D²+E²-4F)

x²+y²+Dx+Ey+F=0 completing the square:
(x+D/2)²+(y+E/2)² = D²/4+E²/4-F.
Center = (-D/2, -E/2).
r = ½√(D²+E²-4F).

Condition for real circle: D²+E²-4F > 0.

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!

6 cm

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

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!

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)!

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.

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)!

Circle ని రెండు points వద్ద కట్ చేసే line

Secant = circle ని exactly two points వద్ద intersect చేసే line.
Secant = Chord ను extend చేసిన line.

Tangent vs Secant:
Tangent → 1 point (touch).
Secant → 2 points (cut through).
గుర్తు: "Sec" = cut చేయడం!

78.5 cm²

r = d/2 = 5 cm.
Area = πr² = 3.14×25 = 78.5 cm².

Both true. External tangents are equal (A). Tangent-radius=90° (R). But R is a separate property, not direct explanation of equal length.
A, R రెండూ సత్యం. కానీ R వేరే గుణం; నేరు వివరణ కాదు.

రెండు radii మరియు arc మధ్య ఉన్న region

Sector = "pie slice" shape.
రెండు radii (OA మరియు OB) మరియు arc AB మధ్య ఉన్న area.

Minor sector = smaller angle, Major sector = larger angle.
గుర్తు: Sector = OA + arc AB + OB అనే boundary తో ఉన్న region.
Segment తో confuse కావద్దు: Segment = chord + arc (radius లేదు)!

Both true. πr²=22/7×49=154. R calculates correctly.
A, R రెండూ సత్యం. వైశాల్యం=154 cm².

AC×BD = AB×CD + BC×DA = 15+24 = 39

Ptolemy: AC×BD = AB×CD + BC×DA.
= 3×5 + 4×6 = 15+24 = 39.

90°

Tangent ⊥ Radius at point of contact → angle = 90°.
ఇది Tangent problems లో foundation!
ఉదా: OT=5 (radius), TP=12 (tangent) → OP=√(25+144)=13 (Pythagoras).

66 cm

C = πd = (22/7)×21 = 22×3 = 66 cm.
లేదా: r=10.5, C=2×(22/7)×10.5=66 cm.

PA = 12 cm

OA⊥PA → PA²=OP²-OA²=169-25=144 → PA=12.
Also PB=12 (equal tangents).

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!

A is true: central angle=2×inscribed angle. R is false: central angle is double the inscribed angle, not equal.
A సత్యం. R అసత్యం: కేంద్ర కోణం=2×పరిధి కోణం, సమానం కాదు.

PC = 9

PA² = PB × PC.
36 = 4 × PC → PC = 9 cm.

x²+y²=16 (Circle center origin, r=4)

PA²=(x-3)²+y², PB²=(x+3)²+y².
PA²+PB²=2x²+18+2y²=50.
2(x²+y²)=32.
x²+y²=16. Circle, r=4!

24 cm

AM² = r² - OM² = 169-25 = 144 → AM=12.
Chord = 2×AM = 24 cm.

60°

∠APC = ½(arc AC + arc BD) = ½(80°+40°) = 60°.

Intersecting Chords Angle Theorem:
Angle = ½ × (sum of intercepted arcs).
గుర్తు: Vertically opposite arcs కూడుతాయి, సగం తీస్తారు!

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.

50°

PA=PB → ∠PAB=∠PBA=65°.
∠APB = 180°-65°-65° = 50°.

180°

∠APB+∠AQB = 180°.
Theorem: ఒక chord యొక్క రెండు వేర్వేరు arcs పై inscribed angles supplementary!

Proof: P minor arc side → ∠APB = (major arc AB)/2.
Q major arc side → ∠AQB = (minor arc AB)/2.
మొత్తం = (major+minor arc)/2 = 360°/2 = 180°.
Cyclic quadrilateral AQBP లో opposite angles!

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.

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).

90° (Right angle)

Thales' Theorem: Diameter పై inscribed angle = 90°.
AB diameter అయినప్పుడు circle పై ఏ point P అయినా:
∠APB = 90°.

Proof: Radius OA=OB=OP → Isosceles triangles.
గుర్తు: "Angle in semicircle is a right angle" — exam లో most important theorem!