Here we will discuss about distance between two points.

* How to find the distance between two given points?*

Or,

* How to find the length of the line segment joining two given points?*

**(A) To find the distance of a given point from the origin:**

Let OX and OYbe the rectangular Cartesian Co-ordinates axes on the plane of reference and the Co-ordinates of a point P on the plane be (x, y). to find the distance of P from the origin O. from P draw PM perpendicular on OX; then , OM = x and PM = y. Now from the right angle triangle OPM we get,

OP² = OM² + PM² = x² + y²

Therefore OP = √(x² + y²) (Since, OP is positive.)

**(B) To find the distance between two points whose rectangular Cartesian co-ordinates are given:**

Let (x₁, y₁) and (x₂, y₂) be the Cartesian co-ordinates of the points P and Q respectively referred to rectangular co-ordinate axes OX and OY. We are to find the distance between the points P and Q. Draw PM and QN perpendiculars from P and Q respectively on OX; then draw PR perpendicular from P on QN.

Clearly, OM = x₁, PM = y₁, ON = x₂ and QN = y₂.

Now, PR = MN = ON - OM = x₂ – x₁

and QR = QN - RN = QN - PM = y₂ – y₁

Therefore, from the right-angled triangle PQR we get,

PQ² = PR² + QR² = (x₂ - x₁)² + ( y₂ - y₁)²

Therefore, PQ = √[(x₂ - x₁)² + (y₂ - y₁)²] (Since, PQ is positive )∙

**Examples on Distance between two Points**

**1.** Find the distance of the point (-5, 12) from the origin.

**Solution:**

We know that, the distance between two given points (x₁, y₁) and (x₂, y₂) is

√{(x₂ - x₁)² + (y₂ - y₁)²}.

The required distance of the point (- 5, 12) from the origin = the distance between the points (- 5, 12) and (0, 0)

= √{(- 5 - 0)² + (12 - 0)²}

= √(25 + 144)

= √169

= 13 units.

**2.** Find the distance between the points (- 2, 5) and (2, 2).

**Solution:**

We know that, the distance between two given points (x₁, y₁) and (x₂, y₂) is

√{(x₂ - x₁)² + (y₂ - y₁)²}.

The required distance between the given points (- 2, 5) and (2, 2)

= √{(2 + 2)² + (2 - 5)²}

= √(16 + 9)

= √25

= 5 units.

**●**** Co-ordinate Geometry**

**What is Co-ordinate Geometry?****Rectangular Cartesian Co-ordinates****Polar Co-ordinates****Relation between Cartesian and Polar Co-Ordinates****Distance between Two given Points****Distance between Two Points in Polar Co-ordinates****Division of Line Segment****: Internal & External****Area of the Triangle Formed by Three co-ordinate Points****Condition of Collinearity of Three Points****Medians of a Triangle are Concurrent****Apollonius' Theorem****Quadrilateral form a Parallelogram****Problems on Distance Between Two Points****Area of a Triangle Given 3 Points****Worksheet on Quadrants****Worksheet on Rectangular – Polar Conversion****Worksheet on Line-Segment Joining the Points****Worksheet on Distance Between Two Points****Worksheet on Distance Between the Polar Co-ordinates****Worksheet on Finding Mid-Point****Worksheet on Division of Line-Segment****Worksheet on Centroid of a Triangle****Worksheet on Area of Co-ordinate Triangle****Worksheet on Collinear Triangle****Worksheet on Area of Polygon****Worksheet on Cartesian Triangle**

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