🗺️ Shapes in Coordinates: Your Treasure Map to Geometry!
Imagine you have a magical treasure map with a grid on it. Every treasure, every landmark, every path can be found using two numbers: how far right (x) and how far up (y). This is Coordinate Geometry — using numbers to describe shapes!
Today, we’ll learn how to find treasure (triangles, circles, special points) using just coordinates. Ready? Let’s go!
🔺 Area of a Triangle Using Coordinates
The Story
Three pirates bury treasure at three spots on their map. How much ground does their triangle cover?
The Magic Formula
If three points are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃):
Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
The | | means “always positive” — area can’t be negative!
Example
Points: A(0, 0), B(4, 0), C(2, 3)
Area = ½ |0(0-3) + 4(3-0) + 2(0-0)|
= ½ |0 + 12 + 0|
= ½ × 12
= 6 square units
Why It Works
Think of it like measuring a pizza slice. The formula counts how much space is inside by using the coordinates as measurements!
➡️ Collinearity Condition
The Story
Three friends standing in a park. Are they in a straight line or forming a triangle?
The Rule
Three points are collinear (on the same line) if the area of the triangle they form is ZERO.
If x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) = 0, points are collinear!
Example
Are A(1, 2), B(3, 4), and C(5, 6) collinear?
= 1(4-6) + 3(6-2) + 5(2-4)
= 1(-2) + 3(4) + 5(-2)
= -2 + 12 - 10
= 0 ✓ Yes! They're on the same line!
Think of it This Way
If three points form a triangle with NO area, they must be squished into a line!
⭕ Circumcenter Coordinates
The Story
Imagine three friends want to stand in a circle, all the same distance from a special center point. That center is the circumcenter!
What is Circumcenter?
- The center of a circle that passes through ALL THREE vertices of a triangle
- It’s equidistant (same distance) from all three corners
Finding the Circumcenter
The circumcenter is where the perpendicular bisectors of the sides meet.
For a triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
graph TD A["Find midpoint of AB"] --> D["Draw line perpendicular to AB at midpoint"] B["Find midpoint of BC"] --> E["Draw line perpendicular to BC at midpoint"] D --> F["Where lines cross = Circumcenter!"] E --> F
Example
For a right triangle at A(0, 0), B(4, 0), C(0, 3):
The circumcenter is at the midpoint of the hypotenuse!
- Hypotenuse BC: midpoint = ((4+0)/2, (0+3)/2) = (2, 1.5)
That’s the circumcenter! The circle with center (2, 1.5) passes through all three points.
🎯 Orthocenter Coordinates
The Story
Drop a perpendicular line from each corner of a triangle to the opposite side. Where do all three lines meet? That’s the orthocenter!
What is Orthocenter?
- The point where all three altitudes meet
- An altitude = a line from a vertex perpendicular to the opposite side
Key Facts
| Triangle Type | Orthocenter Location |
|---|---|
| Acute triangle | Inside the triangle |
| Right triangle | At the right-angle vertex |
| Obtuse triangle | Outside the triangle |
Example
For a right triangle at A(0, 0), B(4, 0), C(0, 3):
The right angle is at A(0, 0). Orthocenter = A(0, 0)
For right triangles, the orthocenter is always at the corner with the 90° angle!
Finding Orthocenter (General Method)
- Find the equation of altitude from vertex A
- Find the equation of altitude from vertex B
- Solve both equations to find intersection point
⚪ Equation of a Circle
The Story
A circle is like a fence around a tree. Every point on the fence is the same distance from the tree (center).
Standard Form
A circle with center (h, k) and radius r:
(x - h)² + (y - k)² = r²
Breaking It Down
- (h, k) = where the center is
- r = how big the circle is (radius)
- Every point (x, y) on the circle satisfies this equation
Example
Circle with center (3, 2) and radius 5:
(x - 3)² + (y - 2)² = 25
Let’s check: Is point (6, 6) on this circle?
(6-3)² + (6-2)² = 9 + 16 = 25 ✓ Yes!
Special Case: Center at Origin
If center is (0, 0):
x² + y² = r²
🔍 Circle from Equation
The Story
Someone gives you a messy equation. Can you figure out where the circle is and how big it is?
General Form
x² + y² + Dx + Ey + F = 0
Converting to Standard Form
Use completing the square!
Example
Find center and radius: x² + y² - 6x + 4y - 12 = 0
Step 1: Group x terms and y terms
(x² - 6x) + (y² + 4y) = 12
Step 2: Complete the square
- For x: take -6, halve it (-3), square it (9)
- For y: take 4, halve it (2), square it (4)
(x² - 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
Step 3: Factor
(x - 3)² + (y + 2)² = 25
Answer: Center = (3, -2), Radius = 5
Quick Reference
| General Form | Find Center | Find Radius |
|---|---|---|
| x² + y² + Dx + Ey + F = 0 | (-D/2, -E/2) | √(D²/4 + E²/4 - F) |
📐 Coordinate Proofs
The Story
Instead of using rulers and protractors, we can prove things about shapes using coordinates!
The Strategy
- Place the shape on a coordinate grid cleverly
- Assign coordinates to vertices
- Calculate using formulas
- Prove what you need!
Common Placements
| Shape | Smart Placement |
|---|---|
| Rectangle | One corner at origin, sides along axes |
| Triangle | One side along x-axis |
| Square | Center at origin |
Example Proof: Diagonals of a Rectangle Bisect Each Other
Setup: Rectangle ABCD
- A = (0, 0)
- B = (a, 0)
- C = (a, b)
- D = (0, b)
Prove: Diagonals AC and BD have the same midpoint
Diagonal AC: from (0, 0) to (a, b)
- Midpoint = ((0+a)/2, (0+b)/2) = (a/2, b/2)
Diagonal BD: from (a, 0) to (0, b)
- Midpoint = ((a+0)/2, (0+b)/2) = (a/2, b/2)
Same point! ✓ Proved!
graph TD A["Step 1: Place shape on grid"] --> B["Step 2: Assign coordinates"] B --> C["Step 3: Use distance/midpoint formulas"] C --> D["Step 4: Show equality or relationship"] D --> E["Proof complete! 🎉"]
Another Example: Midpoint of Hypotenuse
Prove: The midpoint of the hypotenuse of a right triangle is equidistant from all three vertices.
Setup: Right triangle
- A = (0, 0) — right angle
- B = (2a, 0)
- C = (0, 2b)
Midpoint M of hypotenuse BC:
- M = ((2a+0)/2, (0+2b)/2) = (a, b)
Distances:
- MA = √(a² + b²)
- MB = √((a-2a)² + (b-0)²) = √(a² + b²)
- MC = √((a-0)² + (b-2b)²) = √(a² + b²)
All equal! ✓ The midpoint is equidistant from all vertices!
🎯 Summary: Your Coordinate Geometry Toolkit
| Tool | What It Does | Key Formula |
|---|---|---|
| Triangle Area | Find space inside | ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| |
| Collinearity | Check if points are in line | Area = 0 |
| Circumcenter | Circle center through vertices | Perpendicular bisector intersection |
| Orthocenter | Where altitudes meet | Altitude intersection |
| Circle Equation | Describe a circle | (x-h)² + (y-k)² = r² |
| Circle from Equation | Find center & radius | Complete the square |
| Coordinate Proofs | Prove properties | Place → Assign → Calculate → Prove |
🌟 You Did It!
You’ve learned to:
- ✅ Calculate triangle areas with coordinates
- ✅ Check if points are collinear
- ✅ Find circumcenters and orthocenters
- ✅ Write and read circle equations
- ✅ Prove geometric facts with coordinates
Coordinate geometry turns shapes into numbers and numbers into discoveries. You now have the map — go find your treasure! 🗺️✨