Distance Between Two Points: From Axes to Diagonals
How a 2,800-year-old theorem turns coordinate subtraction into the distance between any two points
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A single straight line of bright golden light cuts diagonally across a deep, starry navy sky. From each endpoint of this diagonal, a faint pillar of light extends — one horizontal, one vertical — meeting at a right angle just below the diagonal, forming a perfect right triangle visible only as overlapping beams of light. The image conveys: the distance between two points in a plane is just the hypotenuse of a hidden right triangle. Painterly cinematic illustration in the style of mathematical wonder. Dark background. No text, no labels.
Easy distances first: along an axis or parallel to it
On previous pages you found distances between points whenever the segment between them was parallel to one of the axes. The rule was simple subtraction:
- Two points and on the same horizontal line (same y-coordinate): the distance is .
- Two points and on the same vertical line (same x-coordinate): the distance is .
For example, in Reiaan's room (Page 5), the room door to is horizontal — its width is ft. Easy.
But the segment from to is not parallel to either axis. Subtracting the x-coordinates gives 5, subtracting the y-coordinates gives 4 — neither of these is the distance between the two points. They are something else: the legs of a right triangle whose hypotenuse is the distance we want.
This is the moment Baudhāyana's theorem becomes a tool of coordinate geometry.
Triangle ADM: the worked example
Consider three points in Quadrant I:
Draw segments , , on graph paper. The result is a triangle entirely inside Quadrant I. Triangle ADM is acute-angled (all three angles are less than 90°). Now we ask the obvious question: how long are its sides , , ?
None of these segments is parallel to an axis. Plain subtraction will not give us any of the three side lengths directly. But we can build a right triangle for each side, where the side is the hypotenuse and the legs are parallel to the axes.
AI Generation Prompt
Cartesian plane diagram on green graph paper over a dark background. Two thick green axes meet at the origin. Tick marks at every integer from −1 to +10 on x-axis and from 0 to +6 on y-axis. Three points plotted as small dark dots, each labelled with both name and coordinates: A(3, 4), D(7, 1), M(9, 6). Three green line segments connect them to form an acute-angled triangle ADM in Quadrant I. Style: clean educational mathematics illustration. Dark background, orange accent labels, clean technical illustration style.
Focus on side first. The endpoints are and . To find :
- Drop a vertical line from down to the level of (i.e. to ). Call the foot . Then .
- Now we have three points: , , . Notice that — is directly below and directly left of , so the segment is vertical and the segment is horizontal. Triangle is right-angled at .
- Now compute the two legs by simple subtraction:
- = horizontal distance =
- = vertical distance =
- By Baudhāyana–Pythagoras: , so .
The diagonal distance from to is exactly 5 units. This is the same 3-4-5 right triangle Baudhāyana would have used to lay out a perfect right angle on the ground 2,800 years ago — and it works because of the same theorem he proved.
The pattern is so important it deserves a name. The distance between two points in a plane is the hypotenuse of the right triangle whose legs are parallel to the axes.
Q1.Two points are at and . What is the distance between them?
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). A single straight line of bright golden light cuts diagonally across a deep, starry navy sky. From each endpoint of this diagonal, a faint pillar of light extends — one horizontal, one vertical — meeting at a right angle just below the diagonal, forming a perfect right triangle visible only as overlapping beams of light. The image conveys: the distance between two points in a plane is just the hypotenuse of a hidden right triangle. Painterly cinematic illustration in the style of mathematical wonder. Dark background. No text, no labels.
Easy distances first: along an axis or parallel to it
On previous pages you found distances between points whenever the segment between them was parallel to one of the axes. The rule was simple subtraction:
- Two points and on the same horizontal line (same y-coordinate): the distance is .
- Two points and on the same vertical line (same x-coordinate): the distance is .
For example, in Reiaan's room (Page 5), the room door to is horizontal — its width is ft. Easy.
But the segment from to is not parallel to either axis. Subtracting the x-coordinates gives 5, subtracting the y-coordinates gives 4 — neither of these is the distance between the two points. They are something else: the legs of a right triangle whose hypotenuse is the distance we want.
This is the moment Baudhāyana's theorem becomes a tool of coordinate geometry.
Triangle ADM: the worked example
Consider three points in Quadrant I:
Draw segments , , on graph paper. The result is a triangle entirely inside Quadrant I. Triangle ADM is acute-angled (all three angles are less than 90°). Now we ask the obvious question: how long are its sides , , ?
None of these segments is parallel to an axis. Plain subtraction will not give us any of the three side lengths directly. But we can build a right triangle for each side, where the side is the hypotenuse and the legs are parallel to the axes.
AI Generation Prompt
Cartesian plane diagram on green graph paper over a dark background. Two thick green axes meet at the origin. Tick marks at every integer from −1 to +10 on x-axis and from 0 to +6 on y-axis. Three points plotted as small dark dots, each labelled with both name and coordinates: A(3, 4), D(7, 1), M(9, 6). Three green line segments connect them to form an acute-angled triangle ADM in Quadrant I. Style: clean educational mathematics illustration. Dark background, orange accent labels, clean technical illustration style.
Focus on side first. The endpoints are and . To find :
- Drop a vertical line from down to the level of (i.e. to ). Call the foot . Then .
- Now we have three points: , , . Notice that — is directly below and directly left of , so the segment is vertical and the segment is horizontal. Triangle is right-angled at .
- Now compute the two legs by simple subtraction:
- = horizontal distance =
- = vertical distance =
- By Baudhāyana–Pythagoras: , so .
The diagonal distance from to is exactly 5 units. This is the same 3-4-5 right triangle Baudhāyana would have used to lay out a perfect right angle on the ground 2,800 years ago — and it works because of the same theorem he proved.
The pattern is so important it deserves a name. The distance between two points in a plane is the hypotenuse of the right triangle whose legs are parallel to the axes.
Q1.Two points are at and . What is the distance between them?