Bihar STET Paper 1 - Geometry (ज्यामिति Visual Notes)
Circle Theorems, Cyclic Quadrilateral, Thales (BPT) & Triangle Centers (Pg 147–212)
Weightage: 18–22 Marks
Clean SVG Visual Diagrams
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Type 1: Tangents & Secants (स्पर्श रेखा एवं छेदक रेखा)
Top PriorityVisual Diagram: Tangent-Secant Theorem & Supplementary Tangents
PT² = PA × PB (PB = PA + AB)
∠APB + ∠AOB = 180° (Supplementary)
Core Formulas to Remember:
• Tangent-Secant: PT² = PA × PB (⚠️ Trap: PB poori line hoti hai, sirf AB nahi)
• Intersecting Chords: PA × PB = PC × PD
• Opposite Angle Trick: Tangent Angle + Centre Angle = 180°
• Tangent-Secant: PT² = PA × PB (⚠️ Trap: PB poori line hoti hai, sirf AB nahi)
• Intersecting Chords: PA × PB = PC × PD
• Opposite Angle Trick: Tangent Angle + Centre Angle = 180°
[PYQ Example] Tangent PT = 8 cm, Secant PA = 4 cm. Find chord AB:
PT² = PA × PB
8² = 4 × PB ⇒ 64 = 4 × PB ⇒ PB = 16 cm
Chord AB = PB - PA = 16 - 4 = 12 cm
✔ Answer: 12 cm
8² = 4 × PB ⇒ 64 = 4 × PB ⇒ PB = 16 cm
Chord AB = PB - PA = 16 - 4 = 12 cm
✔ Answer: 12 cm
Type 2: Cyclic Quadrilateral (चक्रीय चतुर्भुज)
180° Rule∠A + ∠C = 180° | ∠B + ∠D = 180° | Exterior Angle = Opposite Interior Angle (θ)
Core Rule: Charo kone circle par honge toh aamne-saamne ke angles ka sum 180° hoga!
[PYQ Example] ∠A = (2x + 4)° and ∠C = (3x + 1)°. Find x:
(2x + 4) + (3x + 1) = 180 ⇒ 5x + 5 = 180 ⇒ 5x = 175
x = 175 / 5 = 35°
✔ Answer: 35°
x = 175 / 5 = 35°
✔ Answer: 35°
Type 3: Thales Theorem (BPT) & Area of Similar Δ
Square RuleAD / DB = AE / EC | AD / AB = DE / BC
⚡ Super Most Repeated Area Theorem:
Do similar triangles ke Area ka ratio unki corresponding sides ke Square (वर्ग) ke barabar hota hai:
Area(Δ1) / Area(Δ2) = (Side 1 / Side 2)² = (Height 1 / Height 2)²
Do similar triangles ke Area ka ratio unki corresponding sides ke Square (वर्ग) ke barabar hota hai:
Area(Δ1) / Area(Δ2) = (Side 1 / Side 2)² = (Height 1 / Height 2)²
[PYQ Example] Side Ratio = 3 : 4, Area 1 = 54 cm². Find Area 2:
54 / Area 2 = (3 / 4)² = 9 / 16
Area 2 = (54 × 16) / 9 = 6 × 16 = 96 cm²
✔ Answer: 96 cm²
Area 2 = (54 × 16) / 9 = 6 × 16 = 96 cm²
✔ Answer: 96 cm²
Type 4: Incenter & Circumcenter Angles (अंतःकेंद्र / परिकेंद्र)
Direct Angles∠BIC = 90° + (∠A / 2)
∠BOC = 2 × ∠A
• Internal Angle Bisector ($I$): ∠BIC = 90° + (∠A / 2)
• External Angle Bisector: ∠BOC = 90° - (∠A / 2)
• Circumcenter ($O$): ∠BOC = 2 × ∠A
• External Angle Bisector: ∠BOC = 90° - (∠A / 2)
• Circumcenter ($O$): ∠BOC = 2 × ∠A
[PYQ Example] In ΔABC, I is incenter. If ∠A = 70°, find ∠BIC:
∠BIC = 90° + (70° / 2) = 90° + 35° = 125°
✔ Answer: 125°
✔ Answer: 125°
