The Rules of Proof
In NCEA, you cannot just look at a shape and say "it looks like a square." You must prove it using the Big Three (Distance, Midpoint, Gradient). Here is the blueprint for proving any quadrilateral.
Parallelogram
How to prove it (Choose One):
- Gradient: Prove opposite sides are parallel ($m_1 = m_2$).
- Midpoint: Prove the diagonals bisect each other (they share the exact same midpoint).
Rectangle
Prove it's a Parallelogram, PLUS (Choose One):
- Gradient: Prove one corner is exactly 90° ($m_1 \times m_2 = -1$).
- Distance: Prove the two diagonals are exactly the same length.
Rhombus
Prove it's a Parallelogram, PLUS (Choose One):
- Distance: Prove two adjacent sides are equal in length.
- Gradient: Prove the diagonals cross at exactly 90°.
Square
The Ultimate Proof:
You must prove it has the properties of BOTH a Rectangle (90° corners) and a Rhombus (equal sides).
Interactive Proof
Use the rules to identify this shape. Calculate the gradients and distances to build an undeniable mathematical proof.
Four coordinates are given: A(0, 0), B(4, 3), C(9, 3), and D(5, 0). Prove what specific shape this forms.
Step 1: Calculate the Gradient of AB.
Step 2: Calculate the Gradient of DC.
Step 3: Calculate Distance of AB.
Step 4: Calculate Distance of BC.
Step 5: Final Conclusion
It is a parallelogram with adjacent sides equal in length. Therefore it is a...
Excellence: Forcing a Shape
Instead of asking you to identify a shape, Excellence questions will give you three points and ask you to find the exact coordinates of the fourth point to force the shape into a specific geometry.
The Parameter Puzzle
Three coordinates are A(0, 0), B(2, 4), and D(-1, 1). We want to place C(k, 5) so that ABCD forms a Parallelogram. Find $k$, then determine if it is a Rhombus.
Step 1: The Diagonals Rule
In a parallelogram, diagonals bisect. So the midpoint of BD must be exactly the same as the midpoint of AC. Calculate the Midpoint of BD.
Step 2: Solve for $k$
The x-coordinate of $M_{AC}$ is $\frac{0 + k}{2}$. Set this equal to the x-coordinate of $M_{BD}$ from Step 1.
Step 3: Distance of AB squared
Step 4: Distance of BC squared
Use $k$ from Step 2 to calculate distance from B to C.
Step 5: Is it a Rhombus?
CS Application: Hitbox Algorithms
The Orthogonal Bounding Box (Hitbox)
1. What is a Hitbox?
In video games, characters have invisible boxes wrapped around them. When a bullet's coordinate enters this box, the game registers a collision (a "hit"). Checking boundaries is mathematically instant if the box is a perfect rectangle.
2. The Rotation Problem
When a character rotates (e.g., a car drifting), the engine uses complex trigonometric matrices to rotate the four corners of the hitbox. However, computer memory has floating-point limits, causing tiny rounding errors.
3. The Geometric Glitch
Because of these rounding errors, the 4 rotated corners might accidentally warp into a weird kite or trapezoid. If this happens, physics glitches occur (like walking through a wall or being shot through cover).
4. The Verification
To prevent this, the engine runs a fast algebraic check every single frame: "Do these 4 rotated coordinates still form a perfect 90° Rectangle?" It calculates gradients and midpoints to verify the shape.
5. The End of the Story (What Happens Next?)
If the test PASSES (It is a Rectangle): The physics engine trusts the coordinates. It proceeds to calculate the super-fast collision math to see if the player gets hit.
If the test FAILS (It is NOT a Rectangle): The engine catches the floating-point glitch! Before allowing the frame to render, it actively forces the coordinates to snap back into a perfect 90° matrix. This prevents players from experiencing broken physics or unfair gameplay.
Engine Verification Test
A game object is rotating. The engine spits out 4 corners: P1(1, 2), P2(3, 6), P3(7, 4), and P4(5, 0). Prove algebraically that this shape is a perfect Rectangle.
Step 1: Check for a 90° Corner (Gradient P1 to P2)
Step 2: Gradient of Adjacent Side (P2 to P3)
Step 3: Verify the Corner
Step 4: Verify it's closed (Diagonals Bisect)
Calculate the Midpoint of diagonal P1-P3. The engine verified P2-P4 has the exact same midpoint!