Terraform Slides TERRAFORM
Mapping by Matrix
12th Grade Advanced Mathematics | Unit: Matrix Applications
The "Parcel B-42" Problem
(2, 14) (18, 16)
You are a Land Surveyor.
Standard formulas like \(\frac{1}{2}bh\) only work for basic triangles. But the real world is irregular.
The Challenge:
Find the EXACT area of this 5-sided plot of land.
Time: 5:00 Minutes for brainstorming strategies...
Triangulating Complex Polygons
Embedded media
Watch closely: How do we transition from a 3-sided triangle to a multi-sided polygon?
The Synthesis Strategy
1
Divide
Split the irregular polygon into non-overlapping triangles using a shared vertex.
2
Determinant
Calculate the area of each triangle using the 3x3 vertex matrix formula.
Area = \(\frac{1}{2} | \text{det}(M) |\)
3
Synthesize
Sum the individual areas to find the Total Parcel Area .
The Vertex Formula
To find the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
Area = \(\frac{1}{2} \left| \text{det} \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|\)
Note: The third column of 1s makes the matrix square and handles the geometric offset.
Project Land Plot
A
Analyze your assigned Plot Coordinate Map.
B
Divide the plot into exactly 4 triangles.
C
Set up and solve four 3x3 determinants.
D
Certify the total area of the parcel.
Team Composition
• 1 Cartographer (Drawing/Dividing)
• 2 Analysts (Matrix Calculations)
• 1 Verifier (Final Summation/Check)
"In surveying, a single digit error in a matrix can cost a company millions in land disputes."
Why Determinants?
Computer Graphics
GPUs use determinants to calculate "face orientation" and lighting for millions of polygons in real-time. This is how 3D video games render solid objects.
GIS & Mapping
Geographic Information Systems use these exact matrix operations to calculate city boundaries, deforestation rates, and property taxes from satellite data.
How would a negative determinant occur in land surveying?
Land Plot Activity Sheet Project Land Plot
Parcel B-42 Area Certification Task
SURVEY NO: 2026-B42
STATUS: UNVERIFIED
Lead Cartographer
Analysis Team
Date
Mission Briefing
The city planning department has provided the following vertex coordinates for an irregular hexagon designated as Parcel B-42 . Your team must calculate the total acreage by triangulating the plot and using matrix determinants. Total area must be precise to 2 decimal places.
Coordinate Registry
Vertex Coordinates (X, Y) A (2, 2) B (8, 3) C (12, 8) D (9, 14) E (3, 12) F (1, 7)
Surveyor's Rough Sketch
Use vertex A (2,2) as your primary triangulation point.
Phase 1: Division
List the three vertices for each of the 4 triangles you will use to calculate the total area.
TRIANGLE 1
A, B, C
TRIANGLE 2
TRIANGLE 3
TRIANGLE 4
Phase 2: Area Certification
Triangle 1: (A, B, C)
\[ M_1 = \begin{pmatrix} 2 & 2 & 1 \\ 8 & 3 & 1 \\ 12 & 8 & 1 \end{pmatrix} \]
Area 1: ________________
Triangle 2: (A, C, D)
\[ M_2 = \begin{pmatrix} 2 & 2 & 1 \\ 12 & 8 & 1 \\ 9 & 14 & 1 \end{pmatrix} \]
Area 2: ________________
Triangle 3: (A, D, E)
\[ M_3 = \begin{pmatrix} 2 & 2 & 1 \\ 9 & 14 & 1 \\ 3 & 12 & 1 \end{pmatrix} \]
Area 3: ________________
Triangle 4: (A, E, F)
\[ M_4 = \begin{pmatrix} 2 & 2 & 1 \\ 3 & 12 & 1 \\ 1 & 7 & 1 \end{pmatrix} \]
Area 4: ________________
Final Land Survey Summary
Sum of Triangle Areas: A1 + A2 + A3 + A4
Synthesis Check: Are any areas negative? (Take Abs Value!)
Total Certified Area (Parcel B-42)
__________ sq units
Parcel Guide Teacher Key TEACHER PARCEL GUIDE
Mapping by Matrix: Parcel B-42 Answer Key & Facilitation
INSTRUCTOR ONLY
Instructional Flow
The Hook (5 min): Prompt students to realize that "Base x Height" is impossible for a hexagon. Students might suggest "cutting it up"—affirm this as Triangulation .Video Synthesis (5 min): Ensure they note the row of 1s in the 3x3 vertex matrix formula.Group Work (25 min): Monitor triangulation. Using vertex A (2,2) as the "pivot" is recommended for simplicity, but any non-overlapping split works.
Common Misconceptions
Forgetting the \(\frac{1}{2}\) factor in the area formula.
Not taking the absolute value of negative determinants.
Misaligning the row of 1s in the 3x3 setup.
Parcel B-42 Certification Key
Triangle 1: A(2,2), B(8,3), C(12,8)
det = [2(3-8) - 2(8-12) + 1(64-36)] = [-10 + 8 + 28] = 26
Area = \(\frac{1}{2}(26) = 13.0\)
Triangle 2: A(2,2), C(12,8), D(9,14)
det = [2(8-14) - 2(12-9) + 1(168-72)] = [-12 - 6 + 96] = 78
Area = \(\frac{1}{2}(78) = 39.0\)
Triangle 3: A(2,2), D(9,14), E(3,12)
det = [2(14-12) - 2(9-3) + 1(108-42)] = [4 - 12 + 66] = 58
Area = \(\frac{1}{2}(58) = 29.0\)
Triangle 4: A(2,2), E(3,12), F(1,7)
det = [2(12-7) - 2(3-1) + 1(21-12)] = [10 - 4 + 9] = 15
Area = \(\frac{1}{2}(15) = 7.5\)
Grand Total Area
88.5 sq units
Formula check: \(13 + 39 + 29 + 7.5 = 88.5\). Ensure students justify why absolute value is necessary (area must be positive).
Discussion / Closure Prompts
1. Coordinate Order Matters:
Ask: "If you swap two rows in your 3x3 matrix, how does it affect the area?" (It flips the sign of the determinant, but the absolute value area remains the same).
2. Efficiency for Computers:
Ask: "Why do programmers use this method instead of Heron's formula?" (Determinants use only addition/multiplication—no square roots—making them computationally 'cheaper' and faster).
