Gradient & Midpoint
Every line segment connecting $A(x_1, y_1)$ and $B(x_2, y_2)$ forms a right-angled triangle. The formulas are simply describing the horizontal and vertical changes.
Gradient ($m$)
Rate of Change
Calculates the steepness. Rise over Run.
Midpoint ($M$)
The Average
Finds the exact center by averaging the coordinates.
Scholarship Proof: Varignon's Theorem
Varignon's Theorem states that connecting the midpoints of any quadrilateral forms a perfect parallelogram. Let's prove the first half: showing that the line connecting the midpoints of AB and BC is exactly parallel to the diagonal AC.
Step 1: Find Midpoint P of side AB.
Step 2: Find Midpoint Q of side BC.
Step 3: Calculate Gradient of PQ.
Step 4: Calculate Gradient of Diagonal AC.
Distance Formula
The distance formula is not a new concept—it is literally just Pythagoras' Theorem ($a^2 + b^2 = c^2$) applied to the coordinate plane.
Distance ($d$)
Data Science: 5D Audience Clustering
In machine learning, algorithms group similar users together by calculating the distance between them in high-dimensional space. The math is the same, we just add more brackets! Calculate the squared distance ($d^2$) between these 3 users based on their daily social media hours to find out who should receive the same targeted ads.
| User | TikTok | YouTube | X (Twitter) | ||
|---|---|---|---|---|---|
| User A | 1 hr | 2 hrs | 3 hrs | 0 hrs | 1 hr |
| User B | 2 hrs | 2 hrs | 1 hr | 1 hr | 1 hr |
| User C | 1 hr | 5 hrs | 3 hrs | 0 hrs | 1 hr |
Step 1: Calculate $d^2$ between User A and User B.
Hint: $(2-1)^2 + (2-2)^2 + (1-3)^2 + \dots$
Step 2: Calculate $d^2$ between User A and User C.
Step 3: Calculate $d^2$ between User B and User C.
Step 4: Conclusion. Which two users are closest together?
The Perpendicular Rule
If two lines intersect at exactly 90°, their gradients are negative reciprocals of each other. If you flip the fraction and change the sign, you have the perpendicular gradient.
The Golden Rule
Or rearranged: $m_2 = -\frac{1}{m_1}$
Concrete Example: Verify the Rule
Let's prove this rule works using two intersecting line segments.
Line 1
Passes through $A(1, 1)$ and $B(2, 3)$.
$m_1 = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2$
Line 2
Passes through $C(0, 3)$ and $D(4, 1)$.
$m_2 = \frac{1 - 3}{4 - 0} = \frac{-2}{4} = -0.5$
Check the perpendicular rule:
$2 \times -0.5 = -1$
The rule holds true!
Parameter Problems
In Scholarship and Excellence-level questions, you aren't asked to find the distance. You are given the distance, but one coordinate is an unknown parameter (like $k$). This requires algebraic solving.
Excellence Walkthrough
"The distance between the points $A(-1, 4)$ and $B(2, k)$ is exactly 5. Find the possible values of $k$."
1. Set up the Distance Formula:
$$ 5 = \sqrt{(2 - (-1))^2 + (k - 4)^2} $$
2. Square both sides to remove the root:
$$ 25 = (3)^2 + (k - 4)^2 $$
$$ 25 = 9 + (k - 4)^2 $$
3. Isolate the algebraic bracket:
$$ 16 = (k - 4)^2 $$
4. Square root both sides (Remember $\pm$!):
$$ \pm 4 = k - 4 $$
5. Solve the two separate equations:
$k = 4 + 4 \implies k = 8$
$k = 4 - 4 \implies k = 0$