Grade 10 Mathematics
Grade 10 · Term 3Mathematics

Analytical Geometry

We place geometric figures on the Cartesian plane and use algebra to derive and apply three key formulas: distance between two points, gradient of a line segment, and midpoint of a segment. These tools let us prove geometric properties algebraically.

7.1 Distance, Gradient & Midpoint7.2 Coordinate Geometry — Quadrilaterals & Lines
Week 4

7.1 Distance, Gradient & Midpoint

  • Derive and apply the distance formula between two points
  • Derive and apply the gradient formula; identify parallel and perpendicular lines
  • Derive and apply the midpoint formula
🌍

Real-World Connection

GPS navigation uses coordinate geometry thousands of times per second. Your phone computes the distance between two coordinates, the bearing (gradient) of each road segment, and midpoints when routing. The same three formulas you learn here run at massive scale inside every navigation app.

Distance between $(x_1;y_1)$ and $(x_2;y_2)$

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Derived from Pythagoras: horizontal change squared plus vertical change squared

Gradient of a line segment

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

m = gradient (slope); undefined for vertical lines (x₁ = x₂)

Property / Rule

Parallel and perpendicular lines

Two lines are parallel if they have equal gradients. Two lines are perpendicular if the product of their gradients is −1.

Parallel: m1=m2Perpendicular: m1×m2=−1\text{Parallel: }m_1=m_2\qquad\text{Perpendicular: }m_1 \times m_2 = -1

Midpoint of segment joining $(x_1;y_1)$ and $(x_2;y_2)$

M=(x1+x22  ;  y1+y22)M = \left(\frac{x_1+x_2}{2}\;;\;\frac{y_1+y_2}{2}\right)

Average of x-coordinates and average of y-coordinates

💡 Tip

Horizontal lines have gradient 0. Vertical lines have undefined gradient. When a gradient is undefined, you cannot use the perpendicular product rule — state instead that the two lines are perpendicular because one is horizontal and one is vertical.

Worked Examples

Worked Example

Apply all three formulas

Problem

Given A(−3; 2)A(-3;\,2) and B(5; −4)B(5;\,-4): find (a) ABAB, (b) gradient of ABAB, (c) midpoint MM of ABAB.

Worked Example

Determine parallel or perpendicular

Problem

Line pp passes through (1;3)(1;3) and (4;7)(4;7). Line qq passes through (0;5)(0;5) and (3;1)(3;1). Are they parallel, perpendicular, or neither?

Worked Example

Find a missing endpoint given the midpoint

Problem

M(2; −1)M(2;\,-1) is the midpoint of PQPQ. P=(−4; 3)P = (-4;\,3). Find QQ.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Find the distance between A(0;0)A(0;0) and B(3;4)B(3;4).
2
L1 · Knowledge2 marks
Find the midpoint of the segment joining P(−2;6)P(-2;6) and Q(4;−2)Q(4;-2).
3
L2 · Routine Procedures4 marks
Find the gradient of the line through (−3;−1)(-3;-1) and (5;3)(5;3), then write the equation of the line.
4
L2 · Routine Procedures2 marks
Line ABAB has gradient 23\frac{2}{3}. What is the gradient of a line perpendicular to ABAB?
5
L3 · Complex Procedures4 marks
Show that triangle ABCABC with A(1;1)A(1;1), B(5;1)B(5;1), C(3;5)C(3;5) is isosceles.
6
L3 · Complex Procedures4 marks
The midpoint of ABAB is M(3;−2)M(3;-2) and A=(7;0)A=(7;0). Find BB and hence find ABAB.
7
L4 · Problem Solving6 marks
Prove that A(−2;1)A(-2;1), B(2;3)B(2;3), C(4;−1)C(4;-1), D(0;−3)D(0;-3) are the vertices of a rectangle.
8
L4 · Problem Solving5 marks
Find the equation of the perpendicular bisector of the segment joining P(1;3)P(1;3) and Q(7;−1)Q(7;-1).
Week 5

7.2 Coordinate Geometry — Quadrilaterals & Lines

  • Represent and classify quadrilaterals on the Cartesian plane using distance and gradient
  • Prove geometric properties using the three coordinate geometry formulas
  • Find equations of lines and use them to solve problems
🌍

Real-World Connection

Town planners lay out roads and plots on a coordinate grid. They check that an intersection is a right angle (perpendicular gradients), that a boundary is exactly bisected (midpoint), and that two parallel roads are equidistant. Every property survey uses these exact calculations.

Property / Rule

Classifying a quadrilateral on the plane

To prove a quadrilateral is a specific type: Parallelogram — two pairs of parallel sides (equal gradients). Rectangle — parallelogram with a right angle (perpendicular gradients). Rhombus — parallelogram with all sides equal (equal distances). Square — rectangle AND rhombus.

Prove: m1=m2 (parallel) or m1m2=−1 (perp)\text{Prove: }m_1=m_2\text{ (parallel) or }m_1 m_2=-1\text{ (perp)}

Equation of a straight line

y−y1=m(x−x1)y - y_1 = m(x - x_1)

m = gradient; $(x_1; y_1)$ = any known point on the line

ℹ️ Note

Strategy for coordinate geometry proofs: (1) State what you need to show. (2) Calculate the relevant gradients and/or distances. (3) Apply the definition. (4) State a conclusion.

Worked Examples

Worked Example

Prove a quadrilateral is a parallelogram

Problem

Show that ABCDABCD with A(0;1)A(0;1), B(3;4)B(3;4), C(7;4)C(7;4), D(4;1)D(4;1) is a parallelogram.

Worked Example

Find the equation of a line through two points

Problem

Find the equation of the line through C(2;−3)C(2;-3) and D(−4;9)D(-4;9).

Worked Example

Prove a quadrilateral is a rhombus

Problem

Prove that PQRSPQRS with P(0;2)P(0;2), Q(3;6)Q(3;6), R(7;6)R(7;6), S(4;2)S(4;2) is a rhombus.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Write the equation of the line with gradient −3-3 passing through (0;4)(0; 4).
2
L1 · Knowledge2 marks
Two lines have gradients m1=34m_1=\frac{3}{4} and m2=−43m_2=-\frac{4}{3}. Are they perpendicular?
3
L2 · Routine Procedures3 marks
Find the equation of the line through (1;5)(1;5) and (3;−1)(3;-1).
4
L2 · Routine Procedures3 marks
Determine whether points A(1;−1)A(1;-1), B(3;3)B(3;3), C(5;7)C(5;7) are collinear.
5
L3 · Complex Procedures6 marks
Show that K(−1;2)K(-1;2), L(2;6)L(2;6), M(6;3)M(6;3), N(3;−1)N(3;-1) form a square.
6
L3 · Complex Procedures3 marks
Line ABAB has equation y=2x−3y = 2x - 3. Find the equation of the line through C(1;4)C(1; 4) parallel to ABAB.
7
L4 · Problem Solving6 marks
The vertices of quadrilateral ABCDABCD are A(−3;0)A(-3;0), B(0;4)B(0;4), C(3;0)C(3;0), D(0;−4)D(0;-4). Prove ABCDABCD is a rhombus and find the equation of diagonal ACAC.
8
L4 · Problem Solving6 marks
A quadrilateral has vertices P(0;0)P(0;0), Q(4;0)Q(4;0), R(5;3)R(5;3), S(1;3)S(1;3). Classify it (with proof) and find the area.

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