Points have an address: (x, y)
Draw two number lines that cross at a right angle. The flat one is the x-axis. The standing one is the y-axis. They meet at the origin O(0, 0).
Every point has two numbers. The first number (x, called the abscissa) says how far to go right or left. The second number (y, called the ordinate) says how far to go up or down. So (3, 2) means 3 right, then 2 up.
Distance formula
Take A(x₁, y₁) and B(x₂, y₂). Walk from A straight across until you are under B. That walk is the x-gap, x₂ − x₁. Then walk straight up to B. That walk is the y-gap, y₂ − y₁. The two walks meet at a right angle, so AB is the hypotenuse of a right triangle.
By Pythagoras: AB = √[(x₂ − x₁)² + (y₂ − y₁)²].
A negative gap does not matter, because squaring makes it positive. Distance of P(x, y) from the origin is √(x² + y²).
Using the distance formula
- Collinear check: three points lie on one line if the two small distances add up to the big one.
- Type of triangle: find all three sides. Equal sides tell you isosceles or equilateral; if a² + b² = c², it is right-angled.
- Equidistant point: if P is equally far from A and B, write PA = PB, square both sides and solve.
Section formula (internal division)
A point P lies on segment AB, between A and B. If AP : PB = m : n, we say P divides AB internally in the ratio m : n.
Think of AB cut into m + n equal pieces. P is m pieces from A. So P moves m/(m + n) of the way from A to B. That gives
P = ( (m·x₂ + n·x₁)/(m + n) , (m·y₂ + n·y₁)/(m + n) )
Why it works: draw small right triangles under AP and PB. They have the same angles, so they are similar, and their gaps are in the ratio m : n. (See Similar triangles.)
Midpoint formula
If m = n, P is the midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2). Just take the average of the x's and of the y's.
Finding the ratio
If the ratio is not known, call it k : 1. Put it in the formula, compare with the given x (or y) and solve for k. To find where the x-axis cuts AB, set y = 0.
Points of trisection
Two points cut AB into three equal parts. They divide AB in 1 : 2 and 2 : 1.
Try it yourself (practical)
At home: Tape two lines on a tiled floor as x and y axes. Stand at tile (1, 1). Put a slipper on tile (4, 5). Count 3 tiles across and 4 tiles up. Now stretch a string straight from you to the slipper and measure it in tile widths: it is 5. You just checked the distance formula.
In the 3D: go to the last step. Set B to (7, 1) and m : n to 1 : 1. Predict P before you look. (Answer: (4, 1), the midpoint.)
Key formulas and definitions
- Distance AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
- Distance from origin = √(x² + y²)
- Section formula (m : n, internal): P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
- Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
- Ratio unknown: take k : 1, then solve for k
Worked examples
1. Find the distance between A(1, 1) and B(4, 5).
x-gap = 4 − 1 = 3. y-gap = 5 − 1 = 4. AB = √(3² + 4²) = √(9 + 16) = √25 = 5 units.
2. Find the distance of P(−6, 8) from the origin.
OP = √[(−6)² + 8²] = √(36 + 64) = √100 = 10 units.
3. Find the midpoint of A(2, −3) and B(6, 7).
x = (2 + 6)/2 = 4. y = (−3 + 7)/2 = 2. Midpoint = (4, 2).
4. Find the point that divides A(1, 1) and B(7, 4) internally in the ratio 2 : 1.
m = 2, n = 1. x = (2×7 + 1×1)/3 = 15/3 = 5. y = (2×4 + 1×1)/3 = 9/3 = 3. P = (5, 3).
5. Are the points A(1, 2), B(3, 6) and C(4, 8) on one straight line?
AB = √(4 + 16) = √20 = 2√5. BC = √(1 + 4) = √5. AC = √(9 + 36) = √45 = 3√5. AB + BC = 2√5 + √5 = 3√5 = AC. So yes, they are collinear.
6. Find the point on the x-axis that is equidistant from A(2, −5) and B(−2, 9).
The point is P(x, 0). PA² = PB²: (x − 2)² + 25 = (x + 2)² + 81. x² − 4x + 4 + 25 = x² + 4x + 4 + 81. −8x = 56, so x = −7. P = (−7, 0).
7. In what ratio does the point (−1, 6) divide the join of A(−3, 10) and B(6, −8)?
Take the ratio k : 1. x = (6k − 3)/(k + 1) = −1. So 6k − 3 = −k − 1, 7k = 2, k = 2/7. Ratio = 2 : 7. Check y: (−8×2 + 10×7)/9 = 54/9 = 6 ✓.
8. Find the points of trisection of the segment joining A(2, −2) and B(−7, 4).
P divides AB in 1 : 2: P = ((−7 + 4)/3, (4 − 4)/3) = (−1, 0). Q divides AB in 2 : 1: Q = ((−14 + 2)/3, (8 − 2)/3) = (−4, 2).
Common mistakes
- Forgetting the square root at the end: √25 = 5, not 25.
- Swapping m and n in the section formula. m goes with the second point (x₂), n with the first point (x₁).
- Writing (x₂ + x₁) instead of (x₂ − x₁) in the distance formula. Distance uses the gap, so subtract.
- Squaring a negative gap wrongly: (−3)² = 9, not −9.