CBSE · Class 10 · Mathematics · Coordinate GeometryDerive the distance formula to find the distance between two points $P(x1, y1)$ and $Q(x2, y2)$ in a Cartesian plane. Using this formula, find the distance between the points $(2, 3)$ and $(4, 1)$.

Introduction to Coordinate Geometry and Distance Formula\nCoordinate geometry establishes a systematic link between algebra and geometry through the use of a Cartesian coordinate system. One of the fundamental building blocks of this branch of mathematics is the ability to determine the exact straight-line distance between any two distinct points located on a coordinate plane.

Step-by-Step Derivation of the Distance Formula

1. Setting up the Plane: Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ be two points plotted in the Cartesian plane.
2. Drawing Perpendicular Lines: Draw perpendicular lines from points $P$ and $Q$ to the $X$-axis, meeting it at points $R$ and $S$ respectively. Thus, the coordinates of $R$ are $(x_1, 0)$ and the coordinates of $S$ are $(x_2, 0)$. The length $RS$ is given by the absolute difference $|x_2 - x_1|$.
3. Constructing a Right-Angled Triangle: Draw a line through point $P$ parallel to the $X$-axis, and draw a line through point $Q$ parallel to the $Y$-axis. Let these two construction lines intersect at a new point $T$.
4. Identifying Coordinates of Point $T$: Since $PT$ is parallel to the $X$-axis, its $Y$-coordinate is the same as the $Y$-coordinate of point $P$, which is $y_1$. Since $QT$ is parallel to the $Y$-axis, its $X$-coordinate is the same as the $X$-coordinate of point $Q$, which is $x_2$. Therefore, the coordinates of point $T$ are $(x_2, y_1)$.
5. Applying Pythagorean Theorem: In the right-angled triangle $\triangle PTQ$, the lengths of the sides are:
   • Base $PT = |x_2 - x_1|$
   • Perpendicular $QT = |y_2 - y_1|$
   • Hypotenuse $PQ = d$ Using Pythagoras theorem ($PQ^2 = PT^2 + QT^2$): $$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$ Taking the square root on both sides: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This is the standard distance formula.

Numerical Application\nNow, let us find the distance between the given points $P(2, 3)$ and $Q(4, 1)$.

• Given: $x_1 = 2, y_1 = 3$ and $x_2 = 4, y_2 = 1$
• Substitute the values into the distance formula: $$d = \sqrt{(4 - 2)^2 + (1 - 3)^2}$| $$d = \sqrt{(2)^2 + (-2)^2}$| $$d = \sqrt{4 + 4}$| $$d = \sqrt{8} = 2\sqrt{2} \text{ units}$$

Conclusion\nThe derived distance formula successfully calculates the length of the line segment joining any two points, and the distance between $(2, 3)$ and $(4, 1)$ is $2\sqrt{2}$ units.