Is it correct to say: Squaring within the distance formula and forgetting the square root?

d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The square root is essential.

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

Point divides P₁P₂ in ratio m:n means AP:PB = m:n. Use (mx₂ + nx₁)/(m + n).

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

Area is always positive. Take |…|. Sign inside indicates orientation, not area.

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

External division: replace n with -n, or use (mx₂ − nx₁)/(m − n).

Is it correct to say: Computing centroid using the 2:1 ratio directly?

Centroid = average of vertices. The 2:1 ratio describes how G sits on each median.

Is it correct to say: Incorrect signs when substituting negative coordinates?

Bracket carefully: (3 − (−1))² = 4² = 16, not 2² = 4.

Coordinate Geometry

Distance formula, section formula, midpoint, area of triangle, centroid, collinearity, slope — Maharashtra SSC Geometry

📐 9 formulas✏️ 3 examples🎯 6 practice⚖️ 5-6 marks🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

💡

Board Exam Tips

→Distance and section formulas — 3-mark ready-made question in Geometry.
→Collinearity of points (area of triangle = 0) is a repeat SSC HOTS.
→Midpoint formula is a special case of section formula (m:n = 1:1).
→Marathi terms: निर्देशक भूमिती (coordinate geometry), मध्यबिंदू (midpoint), केंद्रक (centroid).
→For centroid problems, remember G divides each median in 2:1 ratio (vertex : midpoint).

📐 Formulas(9)

Distance Formula★ Board fav

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

Section Formula (Internal Division)★ Board fav

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

Symbol	Meaning
$m, n$	Ratio in which the point divides the segment
$(x_{1}, y_{1})$	First endpoint
$(x_{2}, y_{2})$	Second endpoint

Midpoint Formula

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

Area of 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 Condition

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

Centroid of Triangle

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

Slope Between Two Points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Section — External Division

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

Distance from Origin

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

✏️ Solved Examples

1Solved Exampleeasy4 steps

Find the distance between (3, 4) and (−1, 1).

Identify coordinates

(x_{1}, y_{1}) = (3, 4), (x_{2}, y_{2}) = (- 1, 1)

2Solved Exampleboard3 steps

Find the coordinates of the point which divides the line segment joining (−1, 3) and (4, −7) in the ratio 3:2 internally.

Identify m:n and endpoints

m : n = 3 : 2; (x_{1}, y_{1}) = (- 1, 3); (x_{2}, y_{2}) = (4, - 7)

3Solved ExampleHOTS6 steps

Prove that the points (0, 0), (3, 4) and (−4, 3) are vertices of an isosceles right-angled triangle.

Label vertices

A (0, 0), B (3, 4), C (- 4, 3)

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Squaring within the distance formula and forgetting the square root
✓d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The square root is essential.
2
Swapping m:n in the section formula
✓Point divides P₁P₂ in ratio m:n means AP:PB = m:n. Use (mx₂ + nx₁)/(m + n).
3
Forgetting absolute value in the area formula
✓Area is always positive. Take |…|. Sign inside indicates orientation, not area.
4
Applying section formula for external division without changing sign
✓External division: replace n with -n, or use (mx₂ − nx₁)/(m − n).
5
Computing centroid using the 2:1 ratio directly
✓Centroid = average of vertices. The 2:1 ratio describes how G sits on each median.
6
Incorrect signs when substituting negative coordinates
✓Bracket carefully: (3 − (−1))² = 4² = 16, not 2² = 4.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the midpoint of the segment joining (2, −3) and (6, 5).
Q2
Find k so that the points (k, 3), (2, −4), (−k + 1, −2) are collinear.
Q3
The centroid of a triangle is (1, 2). Two vertices are (−2, 3) and (4, 5). Find the third vertex.
Q4
Find the point on the x-axis which is equidistant from (2, −5) and (−2, 9).
Q5
If A(1, 2), B(4, 3), C(6, 6), D(3, 5), show that ABCD is a parallelogram.
Q6
Find the area of triangle with vertices (1, 1), (4, 6), (−3, −5).

📝 Notes

Coordinate Geometry

Chapter 5 of Geometry (Part 2) bridges algebra and geometry — every point becomes an ordered pair, every line becomes an equation, and every proof becomes a bit of arithmetic.

SSC angle

Maharashtra SSC Geometry paper usually has:

2-mark: direct distance or midpoint problem.
3-mark: section formula or area of triangle.
HOTS: identify a shape (parallelogram, rhombus, right-angled triangle) from four given points.

Quadrants — sign chart

| Quadrant | x | y | Example | |---|---|---|---| | I | + | + | (3, 4) | | II | − | + | (−3, 4) | | III | − | − | (−3, −4) | | IV | + | − | (3, −4) |

Distance formula intuition

Drop verticals and horizontals from the two points to form a right triangle, apply Pythagoras — that is the whole story.

Section formula — internal vs external

Internal: point lies between A and B ⇒ + signs in the formula.
External: point lies on extended line outside the segment ⇒ − signs.

Identifying shapes from 4 points

Parallelogram: diagonals bisect each other (midpoints coincide).
Rhombus: parallelogram + all sides equal.
Rectangle: parallelogram + diagonals equal.
Square: all sides equal AND diagonals equal.

Board vs CBSE

Both boards use the same formulas. Maharashtra SSC frequently asks proof-style questions ("prove that these points form a rhombus") — memorise the identifying properties.