BIHAR STET PAPER 1: COMPLETE GEOMETRY (ज्यामिति)

Bihar STET Paper 1 - Geometry (ज्यामिति Super Notes)

Circle Tangents, Cyclic Quadrilateral, Thales Theorem & Triangle Similarity (Pages 147–212)

Weightage: 18–22 Marks All 4 High-Yield Types 100% Blogger Ready

Type 1: Circle Tangents & Secants (स्पर्श रेखा एवं छेदक रेखा)

Top Recurring

Master Properties:
• Tangent-Secant Relation: PT² = PA × PB (PB = poori line = PA + AB)
• Intersecting Chords (Andar ya Bahar): PA × PB = PC × PD
• Supplementary Angle Rule: Tangent Angle (∠APB) + Centre Angle (∠AOB) = 180°
• Radius hamesha tangent ke point of contact par 90° banati hai.

[PYQ 1] Tangent-Secant Formula (Chord Length)

Question: From an external point P, tangent PT = 8 cm. A secant line through P cuts circle at A and B. If PA = 4 cm, find the length of chord AB.

PT² = PA × PB
8² = 4 × PB ⇒ 64 = 4 × PB ⇒ PB = 16 cm
Chord AB = PB - PA = 16 - 4 = 12 cm
✔ Correct Answer: 12 cm

[PYQ 2] Tangent & Centre Angle (180° Shortcut)

Question: Tangents PA and PB are drawn to a circle with centre O. If ∠APB = 70°, find ∠AOB.

⚡ Instant Hack: ∠APB + ∠AOB = 180°
∠AOB = 180° - 70° = 110°
✔ Correct Answer: 110°

Type 2: Cyclic Quadrilateral (चक्रीय चतुर्भुज)

Direct Calculation

Golden Rules:
• Opposite Angles Sum: ∠A + ∠C = 180° aur ∠B + ∠D = 180°
• Exterior Angle: Bahar ka exterior angle theek samne wale interior opposite angle ke barabar hota hai.

[PYQ Pattern] Value of x in Cyclic Quadrilateral

Question: In a cyclic quadrilateral ABCD, ∠A = (2x + 4)° and ∠C = (3x + 1)°. Find x.

∠A + ∠C = 180°
(2x + 4) + (3x + 1) = 180 ⇒ 5x + 5 = 180 ⇒ 5x = 175
x = 175 / 5 = 35°
✔ Correct Answer: 35°

Type 3: Thales Theorem (BPT) & Similarity Area Ratio

High Weightage

Core Formulas:
• Thales Theorem / BPT: Agar DE || BC ho, toh AD / DB = AE / EC aur AD / AB = DE / BC
• Similar Triangle 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 = (Median 1 / Median 2)²

[PYQ 1] Similar Triangle Side to Area Ratio

Question: If ΔABC ~ ΔDEF, AB = 3 cm, DE = 4 cm, and Area(ΔABC) = 54 cm², find Area(ΔDEF).

Area(ΔABC) / Area(ΔDEF) = (AB / DE)²
54 / Area(ΔDEF) = (3 / 4)² = 9 / 16
Area(ΔDEF) = (54 × 16) / 9 = 6 × 16 = 96 cm²
✔ Correct Answer: 96 cm²

[PYQ 2] Basic Proportionality Theorem (BPT Side Finding)

Question: In ΔABC, DE || BC. If AD = 2.4 cm, DB = 3.6 cm, and AE = 2 cm, find EC.

AD / DB = AE / EC
2.4 / 3.6 = 2 / EC ⇒ 2 / 3 = 2 / EC ⇒ EC = 3 cm
✔ Correct Answer: 3 cm

Type 4: Incenter & Circumcenter Angles (अंतःकेंद्र / परिकेंद्र)

Direct Trick

Golden Angle Rules:
• Incenter Angle (Internal Bisector I): ∠BIC = 90° + (∠A / 2)
• Excenter Angle (External Bisector): ∠BOC = 90° - (∠A / 2)
• Circumcenter Angle (Parikendra O): ∠BOC = 2 × ∠A (Circle ke center par bana double angle)

[PYQ Pattern] Incenter Angle ∠BIC

Question: In ΔABC, I is the incenter. If ∠A = 70°, find the value of ∠BIC.

∠BIC = 90° + (∠A / 2)
∠BIC = 90° + (70° / 2) = 90° + 35° = 125°
✔ Correct Answer: 125°