Is it correct to say: Writing distance formula as √(x₁² + y₁² − x₂² − y₂²)?

It is √((x₂−x₁)² + (y₂−y₁)²). Difference of coordinates first, then square, then sum.

Is it correct to say: Swapping the weights in the section formula?

m goes with the FARTHER point's coordinates, n with the NEARER. Standard form: (m x₂ + n x₁)/(m+n).

Is it correct to say: Forgetting the absolute value in area formula?

The signed determinant can be negative; area is always its absolute value.

Is it correct to say: Confusing midpoint with centroid formula?

Midpoint: average of TWO points. Centroid: average of THREE vertices of a triangle.

Is it correct to say: Assuming distance can be negative?

Distance is a magnitude ⇒ non-negative. If you get negative, check squaring / signs.

Is it correct to say: Applying the section formula for external division without changing sign?

External division uses (m x₂ − n x₁)/(m − n). Class 10 usually asks internal only, but note the change.

Coordinate Geometry

Distance formula, section formula, midpoint formula and area of a triangle in the Cartesian plane — NCERT Class 10 Maths Ch 7

📐 8 formulas✏️ 3 examples🎯 6 practice⚖️ 4-5 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→Distance formula is the workhorse — write it out at the top of the answer.
→Section formula in ratio m:n internally — a favourite 3-mark question.
→Midpoint = section formula with m:n = 1:1. Memorise once.
→Area of triangle = 0 ⇒ the three points are COLLINEAR — used often in HOTS.
→Plot rough graphs when asked to identify a quadrilateral (parallelogram, rhombus, square).

📐 Formulas(8)

Distance Between Two Points★ Board fav

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Symbol	Meaning
$(x_{1}, y_{1})$	Coordinates of the first point
$(x_{2}, y_{2})$	Coordinates of the second point
$d$	Distance (units)

Distance from Origin

d = x^{2} + y^{2}

Section Formula (Internal Division)★ Board fav

P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

Midpoint Formula★ Board fav

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Centroid of a Triangle

G = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})

Area of a Triangle★ Board fav

Area = \frac{1}{2} x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})

Collinearity Test

x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) = 0

Trisection Points

P_{1} = A + \frac{1}{3} (B - A), P_{2} = A + \frac{2}{3} (B - A)

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the distance between A(3, 4) and B(6, 8).

Apply distance formula

d = (6 - 3)^{2} + (8 - 4)^{2}

2Solved Exampleboard2 steps

Find the coordinates of the point which divides the line segment joining A(2, −2) and B(3, 7) in the ratio 3:2 internally.

Section formula with m = 3, n = 2

x = \frac{3 ( 3 ) + 2 ( 2 )}{3 + 2}, y = \frac{3 ( 7 ) + 2 ( - 2 )}{3 + 2}

3Solved ExampleHOTS4 steps

Prove that the points A(1, 2), B(2, 3) and C(4, 5) are collinear.

Use the collinearity condition (area = 0)

Δ = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Writing distance formula as √(x₁² + y₁² − x₂² − y₂²)
✓It is √((x₂−x₁)² + (y₂−y₁)²). Difference of coordinates first, then square, then sum.
2
Swapping the weights in the section formula
✓m goes with the FARTHER point's coordinates, n with the NEARER. Standard form: (m x₂ + n x₁)/(m+n).
3
Forgetting the absolute value in area formula
✓The signed determinant can be negative; area is always its absolute value.
4
Confusing midpoint with centroid formula
✓Midpoint: average of TWO points. Centroid: average of THREE vertices of a triangle.
5
Assuming distance can be negative
✓Distance is a magnitude ⇒ non-negative. If you get negative, check squaring / signs.
6
Applying the section formula for external division without changing sign
✓External division uses (m x₂ − n x₁)/(m − n). Class 10 usually asks internal only, but note the change.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the distance between (0, 0) and (−3, 4).
Q2
Find the midpoint of the line segment joining (7, −6) and (−3, 4).
Q3
Find the ratio in which the point P(4, 6) divides the segment joining A(1, 3) and B(7, 9).
Q4
If the points (1, 2), (3, k) and (5, 6) are collinear, find k.
Q5
Find the area of the triangle whose vertices are (1, −1), (−4, 6) and (−3, −5).
Q6
Find the coordinates of the centroid of a triangle with vertices A(2, 3), B(−1, 0) and C(5, 6).

📝 Notes

Coordinate Geometry

Coordinate geometry translates plane figures into ordered pairs (x, y), letting us solve geometry problems using algebra.

The three master formulas

Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
Section (m:n internal): ((m x₂ + n x₁)/(m+n), (m y₂ + n y₁)/(m+n))
Area: ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Common quadrilateral tests (using distances)

| Quadrilateral | Check | |---|---| | Parallelogram | Diagonals bisect each other (same midpoint) | | Rhombus | All 4 sides equal | | Rectangle | Opposite sides equal AND diagonals equal | | Square | All 4 sides equal AND diagonals equal |

Ratio of division — quick trick

If P divides AB such that AP:PB = m:n, then:

Farther from A ⇒ m > n.
P coincides with midpoint ⇔ m = n.
P coincides with A ⇔ m = 0; with B ⇔ n = 0.

Collinearity

Three points are collinear if:

Area of triangle formed = 0, OR
Slope of AB = slope of BC.

Quick sanity checks

Distance is symmetric: d(A, B) = d(B, A).
Centroid divides every median in 2:1 from vertex.
Straight line from (x₁, y₁) to (x₂, y₂) has slope (y₂ − y₁)/(x₂ − x₁) if x₂ ≠ x₁.