Coordinate Geometry | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Plot and read coordinates on a Cartesian planePlot dan baca koordinat pada satah Cartesan

Calculate gradient between two pointsKira kecerunan antara dua titik

Find the equation of a straight lineCari persamaan garis lurus

Identify parallel and perpendicular linesKenal pasti garis selari dan serenjang

Find x-intercept and y-intercept from equationsCari pintasan-x dan pintasan-y daripada persamaan

📖 Key Concepts📖 Konsep Utama

Coordinates (Form 2)Koordinat (Tingkatan 2)

A point on the Cartesian plane is written (x, y). The x-axis is horizontal, y-axis is vertical.Titik pada satah Cartesan ditulis (x, y). Paksi-x mengufuk, paksi-y menegak.

Origin = (0, 0) | x-coordinate = horizontal | y-coordinate = vertical

Gradient (Form 2)Kecerunan (Tingkatan 2)

The gradient (slope) measures steepness — the rate of change.Kecerunan mengukur kecuraman — kadar perubahan.

m = (y₂ − y₁) / (x₂ − x₁)
Positive m: line goes up ↗ | Negative m: line goes down ↘
m = 0: horizontal | m = undefined: vertical

Equation of a Straight Line (Form 3)Persamaan Garis Lurus (Tingkatan 3)

y = mx + c where m = gradient, c = y-intercept.y = mx + c di mana m = kecerunan, c = pintasan-y.

Gradient form: y = mx + c
Intercept form: x/a + y/b = 1
m from intercepts: m = −b/a
y-intercept: set x = 0 | x-intercept: set y = 0

Parallel & Perpendicular Lines (Form 3)Garis Selari & Serenjang (Tingkatan 3)

Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1.Garis selari mempunyai kecerunan yang sama. Garis serenjang mempunyai kecerunan yang hasil darabnya −1.

Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1
Perpendicular gradient: m₂ = −1/m₁ (negative reciprocal)

🎮 Interactive: Plot a Line🎮 Interaktif: Plot Garis

Click two points on the grid to draw a line and calculate its gradient!Klik dua titik pada grid untuk melukis garis dan kira kecerunannya!

✏️ Worked Examples✏️ Contoh Penyelesaian

Find the gradient of the line through (2, 3) and (6, 11)Cari kecerunan garis melalui (2, 3) dan (6, 11)

m = (y₂ − y₁) / (x₂ − x₁)

m = (11 − 3) / (6 − 2) = 8 / 4 = 2

m = 2

Find the equation of a line with m = 3 passing through (1, 5)Cari persamaan garis dengan m = 3 melalui (1, 5)

y = mx + c → 5 = 3(1) + c

c = 5 − 3 = 2

y = 3x + 2

Line L has equation y = 2x + 1. Find the equation of the line perpendicular to L through (3, 4).Garis L bersamaan y = 2x + 1. Cari persamaan garis serenjang dengan L melalui (3, 4).

m₁ = 2, so m₂ = −½ (perpendicular: negative reciprocal)

y = −½x + c → 4 = −½(3) + c

c = 4 + 3/2 = 11/2

y = −½x + 5.5

Find the x-intercept and y-intercept of 2x + 3y = 12Cari pintasan-x dan pintasan-y bagi 2x + 3y = 12

y-intercept: set x = 0 → 3y = 12 → y = 4Pintasan-y: tetapkan x = 0 → 3y = 12 → y = 4

x-intercept: set y = 0 → 2x = 12 → x = 6Pintasan-x: tetapkan y = 0 → 2x = 12 → x = 6

y-intercept = 4, x-intercept = 6Pintasan-y = 4, pintasan-x = 6

Find the equation of the line through (−1, 3) and (4, −7)Cari persamaan garis melalui (−1, 3) dan (4, −7)

m = (−7 − 3) / (4 − (−1)) = −10/5 = −2

y = −2x + c → 3 = −2(−1) + c → c = 1

y = −2x + 1

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Mixing up x₁, y₁ order in gradient formulaMengelirukan tertib x₁, y₁ dalam formula kecerunan

m = (y₂ − y₁)/(x₂ − x₁) — y on top, x on bottom. Keep the order consistent! Don't mix (y₂−y₁) with (x₁−x₂).m = (y₂ − y₁)/(x₂ − x₁) — y di atas, x di bawah. Kekalkan tertib konsisten! Jangan campurkan (y₂−y₁) dengan (x₁−x₂).

❌ Confusing parallel vs perpendicular gradientsMengelirukan kecerunan selari vs serenjang

Parallel: m₁ = m₂ (same gradient). Perpendicular: m₁ × m₂ = −1 (negative reciprocal). These are DIFFERENT!Selari: m₁ = m₂ (kecerunan sama). Serenjang: m₁ × m₂ = −1 (resiprokal negatif). Ini BERBEZA!

❌ Confusing x-intercept and y-interceptMengelirukan pintasan-x dan pintasan-y

y-intercept: where line crosses y-axis (set x = 0). x-intercept: where line crosses x-axis (set y = 0). Don't swap them!Pintasan-y: garis memotong paksi-y (tetapkan x = 0). Pintasan-x: garis memotong paksi-x (tetapkan y = 0). Jangan tukar!

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

Gradient of the line through (1, 2) and (4, 8)Kecerunan garis melalui (1, 2) dan (4, 8)

y-intercept of y = 3x − 5Pintasan-y y = 3x − 5

−5

−3

A line parallel to y = 4x + 1 has gradientGaris selari dengan y = 4x + 1 mempunyai kecerunan

−4

−¼

A line perpendicular to y = 2x + 3 has gradientGaris serenjang dengan y = 2x + 3 mempunyai kecerunan

−2

−½

Equation of line with m = −1 through (2, 3)Persamaan garis dengan m = −1 melalui (2, 3)

y = x + 1

y = −x + 5

y = −x + 1

y = x + 5

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

5 marks5 markah

Points A(1, 2) and B(5, k) are on a line with gradient 3.
(a) Find k. [3 marks]
(b) Find the y-intercept of the line. [2 marks]Titik A(1, 2) dan B(5, k) pada garis berkecerunan 3.
(a) Cari k. [3 markah]
(b) Cari pintasan-y garis itu. [2 markah]

Solution (a)Penyelesaian (a)

m = (k − 2) / (5 − 1) = 3

(k − 2) / 4 = 3 → k − 2 = 12

k = 14

k = 14 [3 marks3 markah]

Solution (b)Penyelesaian (b)

Using A(1, 2) and m = 3: y = 3x + c

2 = 3(1) + c → c = −1

y-intercept = −1 [2 marks2 markah]

6 marks6 markah

The line L₁ has equation y = 3x − 4.
(a) Find the y-intercept and x-intercept of L₁. [2 marks]
(b) Line L₂ is perpendicular to L₁ and passes through (6, 1). Find the equation of L₂. [4 marks]Garis L₁ bersamaan y = 3x − 4.
(a) Cari pintasan-y dan pintasan-x L₁. [2 markah]
(b) Garis L₂ serenjang dengan L₁ dan melalui (6, 1). Cari persamaan L₂. [4 markah]

Solution (a)Penyelesaian (a)

y-intercept: c = −4 (from y = 3x − 4)

x-intercept: 0 = 3x − 4 → x = 4/3

y-intercept = −4, x-intercept = 4/3Pintasan-y = −4, pintasan-x = 4/3 [2 marks2 markah]

Solution (b)Penyelesaian (b)

m₁ = 3, so m₂ = −⅓ (perpendicular)

y = −⅓x + c → 1 = −⅓(6) + c

1 = −2 + c → c = 3

y = −⅓x + 3 [4 marks4 markah]

📋 Formula Summary📋 Ringkasan Formula

Coordinate Geometry:Geometri Koordinat:

Gradient:Kecerunan: m = (y₂ − y₁) / (x₂ − x₁)

Equation of a Line:Persamaan Garis:
Gradient form: y = mx + c
Intercept form: x/a + y/b = 1

Intercepts:Pintasan:
y-intercept: set x = 0 | x-intercept: set y = 0

Parallel:Selari: m₁ = m₂
Perpendicular:Serenjang: m₁ × m₂ = −1
Perpendicular gradient:Kecerunan serenjang: m₂ = −1/m₁