Distance Formula

As distance can never be negative, thus we only take the positive square root of the number to find the distance between any two points using the distance formula.

The distance between the points P(x₁, y₁) and Q(x₂, y₂) is

The distance of a point P(x, y) from the origin O(0, 0) is given by

Note:

• To prove that a square is a square, one needs to show that all four sides are equal in length and that the diagonals are also equal in length.
• To prove that a rhombus is a rhombus, one needs to show that all four sides are equal in length.
• To prove that a rectangle is a rectangle, one needs to show that opposite sides are equal in length and that the diagonals are also equal in length.
• To prove that a parallelogram is a parallelogram, one needs to show that opposite sides are equal in length.
• To prove that a figure is a parallelogram but not a rectangle, one needs to show that opposite sides are equal in length but the diagonals are not equal in length.
• To prove that a rhombus is a rhombus but not a square, one needs to show that all sides are equal in length but the diagonals are not equal in length.
• To prove that three points are collinear, one needs to show that the sum of the distance between two pairs of points is equal to the distance between the third pair of points.

Example: Find the distance between the points P(−6, 7) and Q (−1, −5).

Solution:

Here, x₁ = −6, x₂ = −1, y₁ = 7 and y₂ = −5

Substitute the above values in the distance formula.

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Therefore, the distance between the points P(−6, 7) and Q (−1, −5) is 13.

Example: Prove that the points A(1, 7), B(4, 2), C(−1, −1) and D(−4, 4) are the vertices of a square.

Solution:

First, calculate the length of all sides.

AC = BD = 2√17

AB = BC = CD = DA = √34

All sides are equal.

Diagonals are equal.

Hence, it is proved that the points A(1, 7), B(4, 2), C(−1, −1) and D(−4, 4) are the vertices of a square.

CBSE Class 10 Maths Notes Chapter 7 Coordinate Geometry are provided to support student’s education. We believe that students’ learning and development are of the utmost importance, and that’s why we have created these comprehensive notes to help them comprehend the complex subject of Coordinate Geometry better.

The NCERT Class 10 Maths textbook’s Chapter 7 explores the realm of Coordinate Geometry and covers various concepts such as distance formula, section formula, and more. Our notes aim to provide students with a complete summary of the entire chapter, including all essential topics, formulae, and concepts necessary to succeed in their exams.