Geometry & Coordinate Geometry

Scenario

Finding the shortest delivery route for a Zepto rider cutting across a rectangular park using Pythagoras, rather than riding along the roads.

Mental model

Geometry maps physical shapes. Coordinates map them onto a GPS grid.

Pure geometry is about rules that lines and angles follow when they meet. Coordinate geometry takes those rules and plots them on an x-y axis so we can use algebra to find distances and slopes.

Formula

Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2). Slope: m = (y2-y1)/(x2-x1). Pythagoras: a^2 + b^2 = c^2. Area of triangle (coordinates): 0.5 x |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|. Midpoint: ((x1+x2)/2, (y1+y2)/2). Parallel lines: m1 = m2. Perpendicular lines: m1 x m2 = -1. Line equation: y = mx + c (slope-intercept form).

The distance formula is literally Pythagoras applied to the x-y plane. The change in x and change in y are the two legs of the right triangle. The area formula for coordinates is a shortcut for checking if three points are collinear (area = 0 if collinear).

Worked example

Problem: Find the distance between points A(1, 2) and B(4, 6). Also find the midpoint of AB.

1. Distance: d = sqrt((4-1)^2 + (6-2)^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
2. Recognise the 3-4-5 Pythagorean triplet: no calculation even needed.
3. Midpoint: ((1+4)/2, (2+6)/2) = (5/2, 8/2) = (2.5, 4).
4. Sanity-check: Distance from A(1,2) to midpoint(2.5,4) = sqrt(1.5^2 + 2^2) = sqrt(2.25 + 4) = sqrt(6.25) = 2.5. Half of 5. Correct.

Answer: Distance = 5 units, Midpoint = (2.5, 4).

Problem: Find the slope of the line passing through (3, 5) and (7, 13). Is this line parallel or perpendicular to y = 2x + 1?

1. Slope m = (13 - 5) / (7 - 3) = 8 / 4 = 2.
2. The given line y = 2x + 1 also has slope 2.
3. Since m1 = m2 = 2, the lines are parallel (not the same line, just same steepness).
4. Sanity-check for perpendicular: a perpendicular line would have slope -1/2. Our line has slope 2, so NOT perpendicular.

Answer: Slope = 2. The line is parallel to y = 2x + 1.

Shortcuts

• Pythagorean Triplets: Memorize common right triangle sides: (3,4,5), (5,12,13), (8,15,17). Any multiple of these (like 6,8,10) is also a right triangle. (If a triangle has sides 5, 12, and 13, it's instantly a right-angled triangle. No need to calculate squares to check.)
• Collinear Points Check: If 3 points are in a straight line, the area of the triangle formed by them is exactly 0. Or, the slope between AB = slope between BC. (To check if (1,1), (2,2), (3,3) are collinear: slope AB = 1, slope BC = 1. Equal, so collinear.)

Common mistakes

• Confusing Slope and Intercept: In y = mx + c, 'm' is the steepness (slope) and 'c' is where the line hits the y-axis. Do not swap them.
• Assuming lines are parallel just because they look parallel: Lines are only parallel if their slopes are exactly equal (m1 = m2). Never guess by eye from a diagram.

Glossary

collinear

Points that all lie exactly on the same straight line.

slope

A number that measures how steep a straight line is.

parallel

Lines that never meet because they have the exact same steepness.

Recall questions

• What is the equation of a straight line passing through the origin?
• What is the relationship between slopes of two perpendicular lines?
• What do the interior angles of a triangle sum up to?

Questions & answers

Find the distance between the points (2, 3) and (5, 7).

5

Approach: dx = 5-2 = 3. dy = 7-3 = 4. Distance = sqrt(3^2 + 4^2) = sqrt(25) = 5. Recognise the 3-4-5 triplet.

If a triangle has sides 6, 8, and 10, what is its area?

24

Approach: 6, 8, 10 is a Pythagorean triplet (6-8-10 = 2x(3-4-5)). Right triangle. Area = (1/2) x 6 x 8 = 24.

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