Distance Formula

Distance Formula: The distance between two points P(x_1, y_1) and Q(x_2, y_2) in the Cartesian plane is given by:

PQ = sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]

This formula calculates the length of the straight line segment joining the two points. It works for any two points in the plane, regardless of their position (same quadrant, different quadrants, on the axes, or at the origin).

Special Cases:

Important Properties:

• Distance is always non-negative: PQ >= 0.
• Distance is zero if and only if the two points coincide: PQ = 0 implies P = Q.
• Distance is symmetric: PQ = QP (swapping x_1, y_1 with x_2, y_2 gives the same result since we square the differences).
• The order of subtraction does not matter: (x_2 - x_1)^2 = (x_1 - x_2)^2.

Derivation of the Distance Formula:

The Distance Formula is derived directly from the Pythagoras Theorem applied to the Cartesian coordinate system.

Setup: Let P(x_1, y_1) and Q(x_2, y_2) be two points in the Cartesian plane. We want to find the distance PQ.

Step 1: Construction. Draw a horizontal line through P and a vertical line through Q (or vice versa). These lines intersect at a point R, forming a right-angled triangle PQR with the right angle at R.

The coordinates of R are (x_2, y_1) (it shares its x-coordinate with Q and its y-coordinate with P).

Step 2: Find the lengths of the two legs of the right triangle.

The horizontal leg PR has length = |x_2 - x_1| (the horizontal distance between P and R).

The vertical leg QR has length = |y_2 - y_1| (the vertical distance between Q and R).

Step 3: Apply the Pythagoras Theorem.

In the right triangle PQR with hypotenuse PQ:

PQ^2 = PR^2 + QR^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2

Note: We can drop the absolute value signs because squaring makes any negative value positive.

Step 4: Take the positive square root.

PQ = sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]

Since distance is always positive, we take the positive square root.

Conclusion: The distance between P(x_1, y_1) and Q(x_2, y_2) is sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2].

Special Case (Distance from Origin): If P is the origin (0, 0) and Q is (x, y), then: PQ = sqrt[(x - 0)^2 + (y - 0)^2] = sqrt(x^2 + y^2).

Problems involving the Distance Formula in Class 10 fall into several categories:

Type 1: Direct Calculation of Distance

Given two points, find the distance between them. This is the most basic application.

Type 2: Finding Distance from Origin

Find the distance of a given point from the origin O(0, 0).

Type 3: Checking Collinearity

Three points A, B, C are collinear (lie on the same straight line) if and only if AB + BC = AC (or any rearrangement where the two smaller distances add to the largest). Compute all three pairwise distances and check.

Type 4: Classifying Triangles

Given three vertices, compute the three side lengths using the distance formula and determine: (i) is it equilateral (all sides equal)? (ii) is it isosceles (two sides equal)? (iii) is it scalene (no sides equal)? (iv) is it right-angled (check the converse of Pythagoras)?

Type 5: Classifying Quadrilaterals

Given four vertices, compute all four sides and both diagonals to determine whether the quadrilateral is a parallelogram, rectangle, rhombus, square, or none of these.

Type 6: Finding Unknown Coordinates

Given the distance and one point, find the coordinates of another point. This leads to equations involving the distance formula.

Type 7: Equidistant Points

Find points equidistant from two or more given points. This often leads to the equation of a perpendicular bisector or a circle.

Method 1: Direct Distance Computation

Substitute the coordinates directly into d = sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]. Simplify step by step.

Example: Distance between (3, -2) and (-1, 5): d = sqrt[(-1-3)^2 + (5-(-2))^2] = sqrt[16 + 49] = sqrt(65).

Method 2: Collinearity Check

For points A, B, C: compute AB, BC, and AC. If the sum of the two smaller distances equals the largest, the points are collinear.

Example: A(1,1), B(3,3), C(5,5). AB = sqrt(4+4) = 2sqrt(2). BC = sqrt(4+4) = 2sqrt(2). AC = sqrt(16+16) = 4sqrt(2). AB + BC = 4sqrt(2) = AC. Collinear.

Method 3: Triangle Classification

Compute all three side lengths. Check for equality (equilateral/isosceles). Check c^2 vs a^2 + b^2 for right angle.

Method 4: Quadrilateral Classification

Compute all four sides (AB, BC, CD, DA) and both diagonals (AC, BD). Compare to identify the type.

Method 5: Finding Unknown Coordinates

Set up the distance formula equation with the unknown coordinate and the given distance. Square both sides to eliminate the square root, then solve the resulting equation.

Example: Find x if the distance from (x, 3) to (2, 7) is 5. 5 = sqrt[(x-2)^2 + 16]. Squaring: 25 = (x-2)^2 + 16. (x-2)^2 = 9. x-2 = +/-3. x = 5 or x = -1.

Tips:

• Always compute (x_2 - x_1) and (y_2 - y_1) separately before squaring to avoid arithmetic errors.
• The order of points does not matter: d(P,Q) = d(Q,P).
• When comparing distances, you can compare the squares (avoid square roots) to check equality or inequality.
• Leave answers in surd form (e.g., sqrt(65)) unless asked to approximate.

The Distance Formula has widespread applications across mathematics, science, and everyday life.

Navigation and GPS: Modern GPS systems calculate the distance between two locations using a version of the distance formula adapted for spherical coordinates. For short distances on a flat surface, the standard Cartesian distance formula provides accurate results. Navigation apps use this to calculate route distances, estimated travel times, and to find the nearest facilities.

Computer Graphics and Gaming: In video games and computer graphics, the distance formula is used millions of times per frame to calculate distances between objects, detect collisions, render lighting effects (distance from light source to surface), and manage game mechanics like proximity-based interactions.

Robotics: Robots use the distance formula to navigate their environment, measure distances to obstacles, and plan paths. In warehouse automation, robots calculate the shortest distance to their destination using coordinate-based distance computations.

Astronomy: Astronomers use extensions of the distance formula to calculate distances between stars, planets, and galaxies. In a flat (Euclidean) approximation of space, the 3D distance formula d = sqrt[(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2] gives the spatial distance between celestial objects.

Urban Planning: City planners use coordinate geometry and the distance formula to plan road networks, determine optimal locations for schools, hospitals, and fire stations (minimising the maximum distance to any residential area), and design public transportation routes.

Sports Analytics: In cricket, football, and basketball analytics, player positions are tracked using coordinates, and the distance formula calculates pass distances, sprint lengths, and coverage areas. Heat maps of player movement are generated using accumulated distance computations.

Machine Learning: Many machine learning algorithms (such as K-nearest neighbours, K-means clustering, and support vector machines) use the Euclidean distance formula to measure similarity between data points. The distance between feature vectors determines classification and clustering outcomes.