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How does your smartphone map app calculate the exact driving distance between two GPS coordinates or find the midpoint to split a cab fare? Coordinate...
How does your smartphone map app calculate the exact driving distance between two GPS coordinates or find the midpoint to split a cab fare? Coordinate geometry formulas turn geometry into pure algebra.
Why This Chapter Matters
In Class 10 Mathematics, "Coordinate Geometry" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
- Cartesian plane from Class 9.
- Pythagoras theorem.
- Algebraic simplification.
What You Will Learn (Core Objectives)
- Derive and apply the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
- Derive and apply the Section Formula for internal division: $\left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)$.
- Apply the Mid-Point Formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
- Determine collinearity of three points using distance.
- Find the coordinates of special geometric centers (Centroid of triangle: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$).
Chapter Roadmap & Progression
1
1. The Distance Formula
2
2. The Section Formula
3
3. The Midpoint Formula
Complete Concept Guide (100% Curriculum Coverage)
1. The Distance Formula
Using the Pythagorean theorem on the coordinate plane, the distance $d$ between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is: $$\mathbf{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$$ Distance from origin $(0, 0)$ to point $(x, y)$ is simply $\sqrt{x^2 + y^2}$.
2. The Section Formula
The coordinates of a point $P(x, y)$ dividing line segment $AB$ internally in the ratio $m_1 : m_2$ are: $$\mathbf{P(x, y) = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)}$$
3. The Midpoint Formula
When ratio is $1 : 1$, $P$ is the exact midpoint: $$\mathbf{\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$$
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
Find the distance between points $A(2, 3)$ and $B(4, 1)$.
Reveal Answer & Explanation
Answer: $d = \sqrt{(4 - 2)^2 + (1 - 3)^2} = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\text{ units}$.
2*sqrt(2).
2
Find the coordinates of the point that divides the line segment joining $(4, -3)$ and $(8, 5)$ in the ratio $3 : 1$ internally.
Reveal Answer & Explanation
Answer: $x = \frac{3(8) + 1(4)}{3 + 1} = \frac{28}{4} = 7$; $y = \frac{3(5) + 1(-3)}{3 + 1} = \frac{12}{4} = 3$. Point is $(7, 3)$.
Point is (7, 3).
3
Find the ratio in which the $Y$-axis divides the line segment joining $(-4, 5)$ and $(3, -7)$.
Reveal Answer & Explanation
Answer: On $Y$-axis, $x = 0$. Let ratio be $k : 1$. $x = \frac{k(3) + 1(-4)}{k + 1} = 0 \implies 3k - 4 = 0 \implies k = \frac{4}{3}$. Ratio is $4 : 3$.
Ratio is 4:3.
4
If $(1, 2), (4, y), (x, 6),$ and $(3, 5)$ are vertices of a parallelogram taken in order, find $x$ and $y$.
Reveal Answer & Explanation
Answer: Diagonals of a parallelogram bisect each other: midpoints are equal! $\frac{1 + x}{2} = \frac{4 + 3}{2} \implies 1 + x = 7 \implies x = 6$. $\frac{2 + 6}{2} = \frac{y + 5}{2} \implies 8 = y + 5 \implies y = 3$.
x = 6, y = 3.
5
Find the centroid of a triangle whose vertices are $(3, -5), (-7, 4),$ and $(10, -2)$.
Reveal Answer & Explanation
Answer: $G = \left(\frac{3 - 7 + 10}{3}, \frac{-5 + 4 - 2}{3}\right) = \left(\frac{6}{3}, \frac{-3}{3}\right) = (2, -1)$.
Centroid is (2, -1).
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