The Point-Gradient Formula
In Year 11, you used $y = mx + c$. At Excellence/Scholarship level, finding $c$ is a waste of time. Instead, we use the Point-Gradient Formula. It builds the equation of a line instantly if you know just one point and the gradient.
The Superior Formula
Where $m$ is the gradient, and $(x_1, y_1)$ is ANY point on the line.
Why is it better?
Watch how fast we can find the equation of a line with gradient $3$ passing through $(4, 7)$:
$y - y_1 = m(x - x_1)$
$y - 7 = 3(x - 4)$
$y - 7 = 3x - 12$
$y = 3x - 5$
Perpendicular Bisectors
A perpendicular bisector cuts exactly through the middle of a line segment at a 90° angle. Finding its equation is a guaranteed Excellence-level skill that always requires three steps.
Interactive Architecture
Find the equation of the perpendicular bisector of the line segment connecting A(2, 2) and B(8, 6).
Step 1: Find the Midpoint M
Step 2: Find the original Gradient of AB
Round to 2 decimal places if needed.
Step 3: State the Perpendicular Gradient
Step 4: Build the Equation
Use $y - y_1 = m(x - x_1)$ with your Midpoint and $m_{\perp}$. Expand it into $y = mx + c$. What is the $y$-intercept ($c$)?
Data Science: Voronoi Cell Towers
The Big Question: Why find the equation?
If you want to know if a user is closer to Tower 1 or Tower 2, calculating the distance to both is perfectly fine for one user.
But what if a telecom company is tracking 10 million users? Calculating 20 million distance formulas every second will crash their servers! Instead, they use Algorithmic Efficiency.
They calculate the perpendicular boundary line exactly once. Then, they plug the user's GPS coordinates into the boundary equation to check an Inequality ($<$ or $>$). This takes a fraction of a millisecond!
The Algorithm Test
Tower 1 is at $T_1(0, 2)$ and Tower 2 is at $T_2(6, 4)$. Find the boundary equation, then use an inequality to determine which tower a user at $U(4, 5)$ connects to.
Step 1: Find the Midpoint between towers.
Step 2: Calculate the Perpendicular Gradient of the boundary.
First find $m$ of $T_1T_2$, then flip it and change sign.
Step 3: What is the $y$-intercept of the boundary equation?
Build $y = -3x + c$ using the Midpoint.
Step 4: The Inequality Sort
The boundary is $y = -3x + 12$. Plug the user's $x=4$ into $-3x + 12$ to find the boundary $y$. How does the user's actual $y$ (which is $5$) compare to the boundary $y$?
Advanced Collinearity
Points are collinear if they sit on the exact same straight line. This means the gradient between any pair of those points must be identical.
Interactive Excellence Proof
Find the value of $k$ so that the points A(1, 2), B(3, 8), and C(k, 14) are collinear.
Step 1: Calculate the exact Gradient of AB.
Step 2: Solve for $k$
Set up the equation for $m_{BC}$ and make it equal to your answer from Step 1. $\frac{14 - 8}{k - 3} = 3$
Algebraic Proof:
$\frac{6}{k-3} = 3 \implies 6 = 3(k-3) \implies 6 = 3k - 9 \implies 15 = 3k \implies k=5$
Algebra Connection: Infinite Solutions
If two coordinates are collinear with a third, they form one overlapping line. In algebra, when you solve simultaneous equations that describe the exact same line, they intersect everywhere! This creates infinitely many solutions, which occurs when one equation is a direct multiple of the other.
Find the values of $a$ and $b$ so that the system below has infinitely many solutions:
Eq 1: $3x - 4y = 10$
Eq 2: $ax + by = -30$
Step 1: Find the numerical multiplier.
What do you multiply $10$ by to get $-30$?
Step 2: Solve for $a$.
Step 3: Solve for $b$.