The distance formula
Pythagoras on the grid
↳ Across squared plus up squared, then root.
📚 Chapter Learning Kit
Coordinate Geometry · Class 10 Mathematics
English 3 topics 171 leveled MCQs 60 flashcards 3 games Free
Shared by a Veda teacher · Generated with Veda AI · Sep 2026
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💡 Key Idea
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The right triangle
Where it comes from
Draw a horizontal line from one point and a vertical line from the other.
↳ The formula is Pythagoras, nothing more.
A(1, 1) to B(5, 4)
The oldest triangle
↳ A 3-4-5 triangle, drawn on axes.
The section formula
A weighted average of the ends
↳ Note which weight sits on which point.
The error that costs most
The weights are CROSSED
In $m : n$ from A, it is $m$ that multiplies $x_2$ — B's coordinate.
↳ A big m means far from A, so B gets the weight.
The reason behind it
Why crossed?
A large $m$ puts the point FAR from A and CLOSE to B.
↳ It is a balance, not a coincidence.
📖 Smart notes
What you'll study, topic by topic
1
The Distance Formula
The distance between two points is Pythagoras' theorem in disguise: draw the horizontal and vertical legs between them, and the straight line joining them is the hypotenuse. That gives one formula,…
- DISTANCE FORMULA: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
- It comes straight from Pythagoras: the horizontal leg is $x_2 - x_1$ and the vertical leg is $y_2 - y_1$.
- Distance from the ORIGIN: $OP = \sqrt{x^2 + y^2}$ — the same formula with $(0, 0)$.
~40 min · full explanation, examples & memory tricks in the app
2
The Section Formula
A point dividing a segment in the ratio $m : n$ has coordinates that are a weighted average of the endpoints — and the weights are crossed over. The midpoint is simply the case $1 : 1$, which is why its formula is the pl…
- SECTION FORMULA: the point dividing the join of $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m : n$ is…
- The weights are CROSSED: m multiplies the SECOND point's coordinates and n multiplies the FIRST point's.
- MIDPOINT: put $m = n = 1$ to get $\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$.
~45 min · full explanation, examples & memory tricks in the app
3
Putting the Two Formulas to Work
With the distance formula and the section formula in hand, a surprising range of questions becomes routine: proving three points collinear, deciding what kind of quadrilateral four points make, and finding a missing vert…
- COLLINEAR: three points lie on a line exactly when the two shorter distances ADD to the longest.
- Collinearity must be tested with the distances themselves, not their squares — squaring does not respect addition.
- To classify a quadrilateral, compute the four SIDES and the two DIAGONALS, all as squared lengths.
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
171 questions laddered from warm-up to topper-level, each with an explanation. A taste:
The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is
Beginner
A $(x_2-x_1)^2 + (y_2-y_1)^2$
B $\sqrt{(x_2-x_1)^2 - (y_2-y_1)^2}$
C $\sqrt{x_2-x_1} + \sqrt{y_2-y_1}$
D $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Show answer & explanation
$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
The differences are squared, added, then rooted.
The distance formula comes from
Beginner
A the angle sum property
B Pythagoras' theorem
C the section formula
D the quadratic formula
Show answer & explanation
Pythagoras' theorem
The two coordinate gaps are the legs of a right triangle.
The distance of $P(x, y)$ from the origin is
Beginner
A $x + y$
B $\sqrt{x + y}$
C $x^2 + y^2$
D $\sqrt{x^2 + y^2}$
Show answer & explanation
$\sqrt{x^2 + y^2}$
The origin is $(0, 0)$, so the differences are just x and y.
The distance between $(0, 0)$ and $(3, 4)$ is
Beginner
A 7
B 5
C $\sqrt7$
D 25
Show answer & explanation
5
$\sqrt{9 + 16} = 5$.
🃏 Flashcards
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