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32. Plotting Points and Finding Gradient

This section covers plotting points on a Cartesian plane and determining the gradient of a line passing through them.

a. Plotting Points

To plot points on a graph, we use the Cartesian coordinate system, which consists of a horizontal x-axis and a vertical y-axis. The point where they intersect is the origin (0,0).

Given points: A(0,2), B(2,4), C(4,6), D(6,8)

To plot these points:

• Point A (0,2): Start at the origin. Move 0 units along the x-axis (stay at x=0). Move 2 units up along the y-axis. Mark point A.
• Point B (2,4): Start at the origin. Move 2 units to the right along the x-axis. From there, move 4 units up parallel to the y-axis. Mark point B.
• Point C (4,6): Start at the origin. Move 4 units to the right along the x-axis. From there, move 6 units up parallel to the y-axis. Mark point C.
• Point D (6,8): Start at the origin. Move 6 units to the right along the x-axis. From there, move 8 units up parallel to the y-axis. Mark point D.

b. Drawing the Line

Once the points are plotted, use a ruler to draw a straight line that passes through all four points. You will observe that these points are collinear, meaning they lie on the same straight line.

c. Finding the Gradient

The gradient (or slope) of a straight line measures its steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points on the line.

The formula for the gradient m between two points (x₁, y₁) and (x₂, y₂) is:

m = (y₂ − y₁) / (x₂ − x₁)

Let’s use points A(0,2) and B(2,4):

m = (4 − 2) / (2 − 0) = 22 = 1

Let’s verify with points C(4,6) and D(6,8):

m = (8 − 6) / (6 − 4) = 22 = 1

The gradient of the line is 1.

33. Perimeter and Length of a Rectangle

Given: Perimeter P = 36 cm, width w = 8 cm.

Formula for the perimeter of a rectangle: P = 2 × (length + width)

Let l represent the length.

36 = 2(l + 8)

18 = l + 8

l = 10

Length = 10 cm

34. Finding the Mean of a Set of Numbers

Numbers: 10, 15, 20, 25, 30

Sum = 10 + 15 + 20 + 25 + 30 = 100

Count = 5

Mean = 100 ÷ 5 = 20

Mean = 20

35. Rounding to the Nearest Hundred Thousand

Number: 568,349

• Hundred thousands digit = 5.
• Next digit (ten thousands) = 6 (≥5), so round up.
• 5 becomes 6, all following digits become 0.

600,000

36. Area of a Water Trough (Requires Diagram)

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Diagram missing

The water trough diagram is not provided. To calculate the painted outer surface area, we need the shape, dimensions, and which faces were painted. Please provide the diagram or a detailed description.

37. Laptop Hire Purchase Calculation

Cash price = sh.40,000. Hire Purchase (HP) price = 20% more.

20% of 40,000 = 0.20 × 40,000 = 8,000

HP price = 40,000 + 8,000 = 48,000

Deposit = 20,000 → Balance = 48,000 − 20,000 = 28,000

14 equal monthly instalments: 28,000 ÷ 14 = 2,000

(a) Monthly instalment = sh.2,000

Extra paid = HP price − Cash price = 48,000 − 40,000 = 8,000

(b) Extra paid = sh.8,000

38. Area of a Shaded Segment of a Circle

Radius r = 13 cm, central angle θ = 60°.

Step 1: Area of sector

A_sector = (θ / 360°) × π × r²

A_sector = (60 / 360) × π × 169 = 16 × π × 169 = 169π6 cm²

Step 2: Area of triangle

A_triangle = ½ × a × b × sin C

a = 13, b = 13, C = 60°, sin 60° = √3/2 ≈ 0.8660

A_triangle = ½ × 13 × 13 × (√3/2) = (169√3)/4 cm²

Step 3: Shaded segment

A_segment = A_sector − A_triangle = 169π6 − 169√34 cm²

Numerical: π ≈ 3.1416, √3 ≈ 1.73205

169π/6 ≈ 88.488, 169√3/4 ≈ 73.179

Difference ≈ 15.309 cm²

Area ≈ 15.31 cm² (rounded to 2 decimal places)

39. Expression for Total Number of Bananas

Let x = bananas Kamene has.

• Morries: x + 2
• Kamene + Morries = x + (x+2) = 2x + 2
• Mary = (2x+2) − 5 = 2x − 3

Total = x + (x+2) + (2x−3) = 4x − 1

Total bananas = 4x − 1

40. Working with Fractions

Work out: 45 ÷ 1015

1015 = 515

45 ÷ 515 = 45 × 551 = 4×55×51 = 451

451

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Quick Reference Cheatsheet

Mathematics Quick Reference Sheet (Grade 9 CBC)

Algebra

• Linear Equations: ax + b = c → x = (c − b)/a
• Quadratic Formula: x = [−b ± √(b² − 4ac)] / (2a)
• Expressions: Combine like terms, distributive property a(b + c) = ab + ac
• Inequalities: Reverse inequality sign when multiplying/dividing by negative.

Geometry

• Area: Rectangle A = lw, Square A = s², Triangle A = ½ × base × height, Circle A = πr², Sector A = (θ/360°) × πr²
• Perimeter/Circumference: Rectangle P = 2(l + w), Square P = 4s, Circle C = 2πr or πd
• Pythagoras: a² + b² = c² (right triangle)
• Circle Segment: Area = Sector area − Triangle area

Mensuration (3D)

• Volume: Cuboid V = lwh, Cube V = s³, Cylinder V = πr²h, Prism V = Base area × height
• Surface Area: Sum of all faces.

Number System

• Fractions: Common denominator for addition/subtraction; multiply numerator×numerator, denominator×denominator; divide by multiplying by reciprocal.
• Decimals: Rounding – look at the digit to the right.
• Percentages: Convert to fraction or decimal; percentage increase/decrease.
• Mean: Sum ÷ count.

Coordinate Geometry

• Points: (x, y)
• Gradient: m = (y₂ − y₁) / (x₂ − x₁)
• Equation of a line: y = mx + c (c = y‑intercept)

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