COMPLETE 8-IN-1 GEOMETRY (ज्यामिति)

Bihar STET Paper 1 - Complete Geometry (ज्यामिति)

All 8 Core Types • Pure SVG Diagrams • 100% PYQ Decoded (Pages 147–212)

Weightage: 18–22 Marks Complete 8-in-1 Cheatsheet 100% Blogger Ready

Type 1: Tangent-Secant Theorem & Intersecting Chords

Formula #1

PT² = PA × PB (PB = Total Secant = PA + AB)

• Tangent-Secant Relation: PT² = PA × PB
• Two Chords Intersecting (Andar ya Bahar): PA × PB = PC × PD

[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

Type 2: Tangents & Centre Angle (180° Supplementary Rule)

5-Sec Hack

∠APB + ∠AOB = 180°

Rule: Bahar ke point se khinchi gayi do tangents ke beech ka angle (∠APB) aur Centre par bana angle (∠AOB) hamesha 180° (Supplementary) hote hain.

[PYQ Example] Tangents PA & PB have ∠APB = 70°. Find ∠AOB:

∠AOB = 180° - 70° = 110°
✔ Answer: 110°

Type 3: Cyclic Quadrilateral (चक्रीय चतुर्भुज)

180° Rule

∠A + ∠C = 180°  |  ∠B + ∠D = 180°  |  Exterior Angle = Opposite Interior Angle

• Opposite angles sum = 180°
• Exterior angle is equal to the interior opposite angle.

[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°

Type 4: Semi-Circle Angle (90°) & Same Segment Rule

Visual Angle

Diameter par bana angle = 90°

Same Chord ⇒ Equal Angles (θ = θ)

• Semicircle: Angle subtended by a diameter at any point on circle is always 90°.
• Same Segment: Angles in the same segment of a circle are always equal.

[PYQ Example] AB is diameter. C is on circle, ∠CAB = 40°. Find ∠ABC:

∠ACB = 90° (Angle in semicircle)
∠ABC = 180° - (90° + 40°) = 180° - 130° = 50°
✔ Answer: 50°

Type 5: Alternate Segment Theorem (एकांतर वृत्तखंड प्रमेय)

Top Decider

Tangent-Chord Angle (∠BAT) = Opposite Inscribed Angle (∠BCA)

Core Concept: Tangent aur kisi Chord ke beech bana angle theek samne wale opposite circumference ke angle ke barabar hota hai!

[PYQ Example] Tangent at A makes ∠TAB = 60° with chord AB. Find ∠ACB:

By Alternate Segment Theorem:
∠ACB = ∠TAB = 60°
✔ Answer: 60°

Type 6: Thales Theorem (BPT) & Similarity Area Ratio

Area = Side²

AD / DB = AE / EC   |   Area 1 / Area 2 = (Side 1 / Side 2)²

• Thales: AD / DB = AE / EC
• Area Theorem: Area(Δ1) / Area(Δ2) = (Side 1 / Side 2)² = (Height 1 / Height 2)² = (Median 1 / Median 2)²

[PYQ Example] Side Ratio = 3 : 4, Area(ΔABC) = 54 cm². Find Area(ΔDEF):

54 / Area 2 = (3 / 4)² = 9 / 16
Area 2 = (54 × 16) / 9 = 6 × 16 = 96 cm²
✔ Answer: 96 cm²

Type 7: Internal Angle Bisector Theorem (कोण समद्विभाजक)

Side Ratio

AB / AC = BD / DC

Formula: Angle bisector base ko usi ratio me baant-ta hai jis ratio me baki do sides hoti hain:
AB / AC = BD / DC

[PYQ Example] AB = 6 cm, AC = 8 cm, BD = 3 cm. Find DC:

6 / 8 = 3 / DC ⇒ DC = (3 × 8) / 6 = 24 / 6 = 4 cm
✔ Answer: 4 cm

Type 8: Triangle Centers & Common Tangents (DCT / TCT)

Complete Formulas

Triangle Centers Angle Rules:
• Incenter (I): ∠BIC = 90° + (∠A / 2)
• Excenter (External Bisector): ∠BOC = 90° - (∠A / 2)
• Circumcenter (Parikendra O): ∠BOC = 2 × ∠A

Common Tangents Length (Distance = d, Radii = r1, r2):
• Direct Common Tangent (DCT): √[d² - (r1 - r2)²]
• Transverse Common Tangent (TCT): √[d² - (r1 + r2)²]

[PYQ Example] Radii = 8 cm and 3 cm, Distance = 13 cm. Find Direct Common Tangent (DCT):

DCT = √[13² - (8 - 3)²] = √[169 - 25] = √144 = 12 cm
✔ Answer: 12 cm