Geometry and Trigonometry - CSEC Mathematics

Introduction to Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids.

Basic Geometric Concepts

Line segment Line

Types of Angles

Angle Type Description Measure
Acute Less than 90° 0° < θ < 90°
Right Exactly 90° θ = 90°
Obtuse Greater than 90° but less than 180° 90° < θ < 180°
Straight Exactly 180° θ = 180°
Reflex Greater than 180° but less than 360° 180° < θ < 360°

Polygons

A polygon is a closed plane figure with three or more straight sides.

Properties of Common Polygons

Polygon Number of Sides Sum of Interior Angles
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°
n-gon n (n-2) × 180°

Example: Calculating Interior Angles

Find the measure of each interior angle of a regular pentagon.

Solution:

Sum of interior angles = (5-2) × 180° = 540°

Each angle = 540° ÷ 5 = 108°

Triangles

Triangles are three-sided polygons with special properties.

Types of Triangles

Equilateral Isosceles Scalene Right

Pythagoras' Theorem

For a right-angled triangle with hypotenuse c and legs a and b:

a² + b² = c²

Example: Using Pythagoras' Theorem

A right-angled triangle has legs of 3cm and 4cm. Find the length of the hypotenuse.

Solution:

c² = a² + b² = 3² + 4² = 9 + 16 = 25

c = √25 = 5cm

Trigonometry

Trigonometry is the study of relationships between angles and sides of triangles.

Trigonometric Ratios

For a right-angled triangle with angle θ:

θ Hypotenuse Opposite Adjacent

Remember: SOH-CAH-TOA is a mnemonic for remembering the trigonometric ratios:

Special Angles

Angle (θ) sinθ cosθ tanθ
0° 0 1 0
30° 1/2 √3/2 1/√3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 ∞

Circle Geometry

Key Circle Terms

Radius Diameter Arc Chord

Coordinate Geometry

The study of geometry using the coordinate plane.

Distance Formula

The distance between two points (x₁, y₁) and (x₂, y₂) is:

√[(x₂ - x₁)² + (y₂ - y₁)²]

Midpoint Formula

The midpoint between (x₁, y₁) and (x₂, y₂) is:

[(x₁ + x₂)/2, (y₁ + y₂)/2]

Gradient (Slope) of a Line

The gradient m of a line passing through (x₁, y₁) and (x₂, y₂) is:

m = (y₂ - y₁)/(x₂ - x₁)

(x₁,y₁) (x₂,y₂)

Glossary of Terms

Acute Angle
An angle measuring less than 90 degrees.
Chord
A straight line connecting two points on a curve, especially on a circle's circumference.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Polygon
A closed plane figure with three or more straight sides.
Pythagoras' Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Radius
A straight line from the center to the circumference of a circle or sphere.
Sector
A portion of a circle enclosed by two radii and an arc.
Tangent
A straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point.
Trigonometry
The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

Self-Assessment Questions

  1. Calculate the size of each interior angle of a regular hexagon.

    2. Sum of interior angles = (6-2) × 180° = 720°

    Each angle = 720° ÷ 6 = 120°

  2. A right-angled triangle has one leg measuring 5cm and hypotenuse measuring 13cm. Find the length of the other leg.

    3. Using Pythagoras' theorem: a² + 5² = 13²

    a² = 169 - 25 = 144

    a = √144 = 12cm

  3. Find the length of the hypotenuse of a right-angled triangle where the other two sides are both 1 unit long.

    4. h² = 1² + 1² = 2

    h = √2 units

  4. Calculate the area of a circle with radius 7cm (take π = 22/7).

    5. Area = πr² = (22/7) × 7 × 7 = 154cm²

  5. Find the distance between points (2, 3) and (5, 7) on the coordinate plane.

    6. Distance = √[(5-2)² + (7-3)²] = √[9 + 16] = √25 = 5 units

  6. Find the midpoint between (1, 2) and (9, 6).

    7. Midpoint = [(1+9)/2, (2+6)/2] = [5, 4]

  7. Calculate the gradient of the line passing through points (1, 1) and (3, 9).

    8. Gradient = (9-1)/(3-1) = 8/2 = 4

  8. A ladder 5m long leans against a wall with its foot 3m from the wall. How high up the wall does the ladder reach?

    9. Using Pythagoras' theorem: h² + 3² = 5²

    h² = 25 - 9 = 16

    h = √16 = 4m

  9. Find the value of sin60° + cos30°.

    10. sin60° = √3/2, cos30° = √3/2

    √3/2 + √3/2 = √3

  10. Calculate the circumference of a circle with diameter 14cm (take π = 22/7).

    11. Circumference = πd = (22/7) × 14 = 44cm