Geometry and Trigonometry

Overview

Geometry and Trigonometry form the backbone of the Mathematics section in TN TET Paper II. This topic carries significant weightage, typically accounting for 8-12 questions across both content knowledge and pedagogical application. The scope covers triangles (properties, congruence, similarity), circles (theorems, tangents, chords), coordinate geometry (distance, section formula, straight lines), and basic trigonometry (ratios, identities, heights and distances).

For classes 6-8, students transition from intuitive geometry to formal proofs and from concrete shapes to abstract coordinate systems. As a TET aspirant, you must demonstrate both mathematical competence and awareness of how children develop geometric reasoning. Expect direct calculation problems, theorem applications, and questions linking geometry concepts to classroom teaching strategies.

Mastery here requires memorising key formulas, understanding theorem conditions, and recognising common problem patterns. This topic also connects strongly to mensuration, so clarity on basic properties will help across multiple areas.

Key Concepts

• **Triangle Classification**: By sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). The angle sum property states that interior angles always total 180°.

• **Congruence Criteria**: Two triangles are congruent if they satisfy SSS, SAS, ASA, AAS, or RHS (for right triangles). Congruent triangles have equal corresponding sides and angles.

• **Similarity Criteria**: Triangles are similar if corresponding angles are equal (AA criterion) or sides are proportional (SSS or SAS similarity). Similar triangles have proportional sides but not necessarily equal.

• **Circle Fundamentals**: A circle is the locus of points equidistant from a centre. Key parts include radius, diameter, chord, arc, sector, and segment. The diameter is the longest chord.

• **Circle Theorems**: Angle in a semicircle is 90°. Angles in the same segment are equal. The perpendicular from the centre bisects the chord. Tangent is perpendicular to the radius at the point of contact.

• **Coordinate Geometry Basics**: The Cartesian plane uses ordered pairs (x, y). The x-axis is horizontal, y-axis is vertical. Quadrants are numbered anticlockwise from the top-right.

• **Trigonometric Ratios**: For a right triangle with angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Reciprocals are cosec, sec, and cot.

• **Standard Angle Values**: Memorise values for 0°, 30°, 45°, 60°, and 90°. These appear repeatedly in calculations.

Formulas / Key Facts

### Triangle Formulas

• Angle sum: A + B + C = 180°
• Exterior angle = Sum of two interior opposite angles
• Area = (1/2) × base × height
• Pythagoras theorem (right triangle): a² + b² = c² where c is hypotenuse
• Basic Proportionality Theorem: A line parallel to one side divides the other two sides proportionally

### Circle Formulas

• Circumference = 2πr
• Area = πr²
• Arc length = (θ/360°) × 2πr
• Sector area = (θ/360°) × πr²
• Length of tangent from external point: √(d² − r²) where d is distance from centre

### Coordinate Geometry Formulas

• Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]
• Section formula (internal division in ratio m:n) = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)]
• Slope of line = (y₂ − y₁)/(x₂ − x₁)
• Equation of line: y − y₁ = m(x − x₁)

### Trigonometry Formulas

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• tan θ = sin θ / cos θ

### Standard Trigonometric Values | Angle | 0° | 30° | 45° | 60° | 90° | |-------|-----|------|------|------|-----| | sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan | 0 | 1/√3 | 1 | √3 | undefined |

Worked Examples

**Example 1: Triangle Similarity** In triangle ABC, DE is parallel to BC where D lies on AB and E lies on AC. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, find EC.

*Solution*: By Basic Proportionality Theorem: AD/DB = AE/EC 4/6 = 5/EC EC = (5 × 6)/4 = 30/4 = 7.5 cm

**Example 2: Distance Formula** Find the distance between points A(3, 4) and B(−1, 1).

*Solution*: Distance = √[(−1 − 3)² + (1 − 4)²] = √[(−4)² + (−3)²] = √[16 + 9] = √25 = 5 units

**Example 3: Trigonometry Application** If sin θ = 3/5, find cos θ and tan θ (θ is acute).

*Solution*: Using sin²θ + cos²θ = 1: (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 cos²θ = 16/25 cos θ = 4/5 (positive since θ is acute)

tan θ = sin θ / cos θ = (3/5)/(4/5) = 3/4

**Example 4: Circle Theorem** A chord is 8 cm long and its distance from the centre is 3 cm. Find the radius.

*Solution*: Perpendicular from centre bisects the chord. Half-chord = 4 cm, distance = 3 cm Using Pythagoras: r² = 4² + 3² = 16 + 9 = 25 r = 5 cm

Common Mistakes

• **Confusing congruence with similarity** → Congruence means identical in shape AND size (use SSS, SAS, ASA, AAS, RHS). Similarity means same shape but proportional size (use AA, SSS ratio, SAS ratio). Check whether the question asks for equal sides or proportional sides.

• **Applying Pythagoras to non-right triangles** → Pythagoras theorem applies ONLY to right-angled triangles. For other triangles, use the cosine rule or other methods. Always verify the triangle has a 90° angle first.

• **Sign errors in coordinate geometry** → When points have negative coordinates, students often drop the minus sign during subtraction. Write out (x₂ − x₁) fully before calculating, especially with negatives like (−3 − 5) = −8, not −2.

• **Mixing up trigonometric ratios** → Students confuse which side is opposite and which is adjacent. The "opposite" side is across from the angle; the "adjacent" side touches the angle (but is not the hypotenuse). Use the mnemonic SOH-CAH-TOA.

• **Using degrees in identity problems** → The identities sin²θ + cos²θ = 1 work for any angle, but students sometimes substitute degree values incorrectly. When given sin 30° = 1/2, use the fraction 1/2 in calculations, not "30".

Quick Reference

• Triangle angle sum = 180°; quadrilateral = 360°
• For congruence, both shape and size match; for similarity, only shape matches
• Distance formula: √[(x₂ − x₁)² + (y₂ − y₁)²]
• sin²θ + cos²θ = 1 — the most frequently used identity
• Tangent ⊥ radius at point of contact
• sin 30° = cos 60° = 1/2; sin 45° = cos 45° = 1/√2; sin 60° = cos 30° = √3/2